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host.h File Reference

Host. More...

Macros

Configuration options
Defined if multi-threading support is disabled. More...

#define VL_DISABLE_SSE2
Defined if SSE2 support if disabled. More...

#define VL_DISABLE_OPENMP
Defined if OpenMP support is disabled. More...

Defining functions
#define VL_EXPORT
Declares a DLL exported symbol. More...

#define VL_INLINE
Adds appropriate inline function qualifier. More...

C preprocessor helper macros
#define VL_STRINGIFY(x)   # x
Convert the argument to a string. More...

#define VL_XSTRINGIFY(x)   VL_STRINGIFY(x)
Expand and then convert the argument to a string. More...

#define VL_CAT(x, y)   x ## y
Concatenate two arguments into a lexical unit. More...

#define VL_XCAT(x, y)   VL_CAT(x,y)
Expand and then concatenate two arguments into a lexical unit. More...

#define VL_XCAT3(x, y, z)   VL_XCAT(VL_XCAT(x,y),z)
Expand and then concatenate three arguments into a lexical unit. More...

#define VL_XCAT4(x, y, z, u)   VL_XCAT(VL_XCAT3(x,y,z),u)
Expand and then concatenate four arguments into a lexical unit. More...

#define VL_XCAT5(x, y, z, u, v)   VL_XCAT(VL_XCAT4(x,y,z,u),v)
Expand and then concatenate five arguments into a lexical unit. More...

#define VL_YESNO(x)   ((x)?"yes":"no")
Convert a boolean to "yes" or "no" strings. More...

Identifying the host operating system
#define VL_OS_LINUX   1
Defined if the host operating system is Linux.

#define VL_OS_MACOSX   1
Defined if the host operating system is Mac OS X.

#define VL_OS_WIN   1
Defined if the host operating system is Windows (32 or 64)

#define VL_OS_WIN64   1
Defined if the host operating system is Windows-64.

Identifying the host threading library
Defiend if the host uses POISX threads. More...

Defined if the host uses Windows threads. More...

Identifying the host compiler
#define VL_COMPILER_GNUC
Defined if the host compiler is GNU C. More...

#define VL_COMPILER_MSC   _MSC_VER
Defined if the host compiler is Microsoft Visual C++. More...

#define VL_COMPILER_LCC   1
Defined if the host compiler is LCC. More...

Identifying the host CPU architecture
#define VL_ARCH_IX86   300
Defined if the host CPU is of the Intel x86 family. More...

#define VL_ARCH_IA64
Defined if the host CPU is of the Intel Architecture-64 family. More...

Identifying the host data model
#define VL_COMPILER_LLP64
Defined if the host compiler data model is LLP64. More...

#define VL_COMPILER_LP64
Defined if the host compiler data model is LP64. More...

#define VL_COMPILER_ILP32
Defined if the host compiler data model is ILP32. More...

Identifying the host endianness
#define VL_ARCH_LITTLE_ENDIAN
Defined if the host CPU is little endian. More...

#define VL_ARCH_BIG_ENDIAN
Defined if the host CPU is big endian. More...

Creating integer constants
#define VL_INT8_C(x)   x
Create an integer constant of the specified width and sign. More...

#define VL_INT16_C(x)   x
Create an integer constant of the specified width and sign. More...

#define VL_INT32_C(x)   x
Create an integer constant of the specified width and sign. More...

#define VL_INT64_C(x)   x ## L
Create an integer constant of the specified width and sign. More...

#define VL_UINT8_C(x)   x
Create an integer constant of the specified width and sign. More...

#define VL_UINT16_C(x)   x
Create an integer constant of the specified width and sign. More...

#define VL_UINT32_C(x)   x ## U
Create an integer constant of the specified width and sign. More...

#define VL_UINT64_C(x)   x ## UL
Create an integer constant of the specified width and sign. More...

Printing the atomic data types
#define VL_FL_INT64   "I64"
printf length flag for vl_int64 and vl_uint64.

#define VL_FL_INT32   ""
printf length flag for vl_int32 and vl_uint32.

#define VL_FL_INT16   "h"
printf length flag for vl_int16 and vl_uint16.

#define VL_FL_INT8   "hh"
printf length flag for vl_int8 and vl_uint8.

#define VL_FL_INDEX   VL_FL_INT64
printf length flag for vl_index and vl_uindex

#define VL_FL_INDEX   VL_FL_INT32
printf length flag for vl_index and vl_uindex

#define VL_FMT_INDEX   VL_FL_INDEX "d"
printf flag for vl_index

#define VL_FMT_INTPTR   VL_FMT_INDEX
printf flag for vl_intptr

#define VL_FMT_UINDEX   VL_FL_INDEX "u"
printf flag for vl_uindex

#define VL_FMT_SIZE   VL_FMT_UINDEX
printf flag for vl_size

#define VL_FMT_UINTPTR   VL_FMT_UINDEX
printf flag for vl_uintptr

Atomic data types limits
#define VL_BIG_INT   0x7FFFFFFFL
Largest integer (math constant)

#define VL_SMALL_INT   (- VL_BIG_INT - 1)
Smallest integer (math constant)

#define VL_BIG_UINT   0xFFFFFFFFUL
Largest unsigned integer (math constant)

Functions

char * vl_static_configuration_to_string_copy ()
Human readable static library configuration. More...

Endianness detection and conversion
void vl_swap_host_big_endianness_8 (void *dst, void *src)
Host <-> big endian transformation for 8-bytes value. More...

void vl_swap_host_big_endianness_4 (void *dst, void *src)
Host <-> big endian transformation for 4-bytes value. More...

void vl_swap_host_big_endianness_2 (void *dst, void *src)
Host <-> big endian transformation for 2-bytes value. More...

Obtaining host info at run time
void _vl_x86cpu_info_init (VlX86CpuInfo *self)

char * _vl_x86cpu_info_to_string_copy (VlX86CpuInfo const *self)

Atomic data types

#define VL_TRUE   1
true (1) constant

#define VL_FALSE   0
false (0) constant

typedef long long vl_int64
Signed 64-bit integer.

typedef int vl_int32
Signed 32-bit integer.

typedef short vl_int16
Signed 16-bit integer.

typedef char vl_int8
Signed 8-bit integer.

typedef long long unsigned vl_uint64
Unsigned 64-bit integer.

typedef int unsigned vl_uint32
Unsigned 32-bit integer.

typedef short unsigned vl_uint16
Unsigned 16-bit integer.

typedef char unsigned vl_uint8
Unsigned 8-bit integer.

typedef int vl_int
Same as int.

typedef unsigned int vl_uint
Same as unsigned int.

typedef int vl_bool
Boolean.

typedef vl_int64 vl_intptr
Integer holding a pointer.

typedef vl_uint64 vl_uintptr
Unsigned integer holding a pointer.

typedef vl_uint64 vl_size
Unsigned integer holding the size of a memory block.

typedef vl_int64 vl_index
Signed version of vl_size and vl_uindex.

typedef vl_uint64 vl_uindex
Same as vl_size.

Detailed Description

Portability features

◆ VL_ARCH_BIG_ENDIAN

 #define VL_ARCH_BIG_ENDIAN
Endianness

◆ VL_ARCH_IA64

 #define VL_ARCH_IA64

◆ VL_ARCH_IX86

 #define VL_ARCH_IX86   300

◆ VL_ARCH_LITTLE_ENDIAN

 #define VL_ARCH_LITTLE_ENDIAN
Endianness

◆ VL_CAT

 #define VL_CAT ( x, y ) x ## y
Parameters
 x first argument to be concatenated. y second argument to be concatenated.

This macro concatenates its arguments into a single lexical unit by means of the ## preprocessor operator. Notice that arguments concatenated by ## are not pre-expanded by the C preprocessor. To macro-expand the arguments and then concatenate them,use VL_XCAT.

VL_XCAT

◆ VL_COMPILER_GNUC

 #define VL_COMPILER_GNUC
Value:
(__GNUC__ * 10000 \
+ __GNUC_MINOR__ * 100)

This macro is defined if the compiler is GNUC. Its value is calculated as

10000 * MAJOR + 100 * MINOR + PATCHLEVEL
Host compiler

◆ VL_COMPILER_ILP32

 #define VL_COMPILER_ILP32
Data models

◆ VL_COMPILER_LCC

 #define VL_COMPILER_LCC   1
Deprecated:
The LCC is not supported anymore.
Host compiler

◆ VL_COMPILER_LLP64

 #define VL_COMPILER_LLP64
Data models

◆ VL_COMPILER_LP64

 #define VL_COMPILER_LP64
Data models

◆ VL_COMPILER_MSC

 #define VL_COMPILER_MSC   _MSC_VER
Host compiler

◆ VL_DISABLE_OPENMP

 #define VL_DISABLE_OPENMP

Define this symbol during compilation of the library and linking to another project to disable VLFeat OpenMP support.

◆ VL_DISABLE_SSE2

 #define VL_DISABLE_SSE2

Define this symbol during compliation of the library and linking to another project to disable VLFeat SSE2 support.

Define this symbol during compilation of the library and linking to another project to disable VLFeat multi-threading support.

◆ VL_EXPORT

 #define VL_EXPORT

◆ VL_INLINE

 #define VL_INLINE

◆ VL_INT16_C

 #define VL_INT16_C ( x ) x
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_INT32_C

 #define VL_INT32_C ( x ) x
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_INT64_C

 #define VL_INT64_C ( x ) x ## L
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_INT8_C

 #define VL_INT8_C ( x ) x
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_STRINGIFY

 #define VL_STRINGIFY ( x ) # x
Parameters
 x value to be stringified.

This macro stringifies the argument x by means of the # prerpocessor operator.

The standard C preprocessor does not prescan arguments which are stringified, so

#define A B
char const * str = VL_STRINGIFY(A) ;

initializes str with a pointer to the string "A", which mihgt be unexpected. To fix this issue, you can use VL_XSTRINGIFY.

VL_XSTRINGIFY

◆ VL_UINT16_C

 #define VL_UINT16_C ( x ) x
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_UINT32_C

 #define VL_UINT32_C ( x ) x ## U
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_UINT64_C

 #define VL_UINT64_C ( x ) x ## UL
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_UINT8_C

 #define VL_UINT8_C ( x ) x
Parameters
 x integer constant.
Returns
x with the correct suffix for the given sign and size. The suffix used depends on the Data models.
Example:
The macro VL_INT64_C(1234) is expanded as 123L in a LP64 system and as 123LL in a LLP64 system.

◆ VL_XCAT

 #define VL_XCAT ( x, y ) VL_CAT(x,y)
Parameters
 x first argument to be concatenated. y second argument to be concatenated.

This macro is the same as VL_CAT, except that the arguments are macro expanded before being concatenated.

VL_CAT

◆ VL_XCAT3

 #define VL_XCAT3 ( x, y, z ) VL_XCAT(VL_XCAT(x,y),z)
Parameters
 x first argument to be concatenated. y second argument to be concatenated. z third argument to be concatenated.

This macro is the same as VL_XCAT, except that it has three arguments.

VL_XCAT

◆ VL_XCAT4

 #define VL_XCAT4 ( x, y, z, u ) VL_XCAT(VL_XCAT3(x,y,z),u)
Parameters
 x first argument to be concatenated. y second argument to be concatenated. z third argument to be concatenated. u fourth argument to be concatenated.

This macro is the same as VL_XCAT, except that it has four arguments.

VL_XCAT

◆ VL_XCAT5

 #define VL_XCAT5 ( x, y, z, u, v ) VL_XCAT(VL_XCAT4(x,y,z,u),v)
Parameters
 x first argument to be concatenated. y second argument to be concatenated. z third argument to be concatenated. u fourth argument to be concatenated. v fifth argument to be concatenated.

This macro is the same as VL_XCAT, except that it has five arguments.

VL_XCAT

◆ VL_XSTRINGIFY

 #define VL_XSTRINGIFY ( x ) VL_STRINGIFY(x)
Parameters
 x value to be macro-expanded and converted.

This macro macro-expands the argument x and stringifies the result of the expansion. For instance

#define A B
char const * str = VL_STRINGIFY(A) ;

initializes str with a pointer to the string "B".

VL_STRINGIFY

◆ VL_YESNO

 #define VL_YESNO ( x ) ((x)?"yes":"no")
Parameters
 x boolean to convert.

A pointer to either the string "yes" (if x is true) or the string "no".

Example
VL_PRINTF("Is x true? %s.", VL_YESNO(x))

◆ vl_static_configuration_to_string_copy()

 char* vl_static_configuration_to_string_copy ( )
Returns
a new string with the static configuration.

The string includes information about the compiler, the host, and other static configuration parameters. The string must be released by vl_free.

◆ vl_swap_host_big_endianness_2()

 void vl_swap_host_big_endianness_2 ( void * dst, void * src )
inline
Parameters
 dst destination 2-byte buffer. src source 2-byte bufffer.
Endianness.

◆ vl_swap_host_big_endianness_4()

 void vl_swap_host_big_endianness_4 ( void * dst, void * src )
inline
Parameters
 dst destination 4-byte buffer. src source 4-byte bufffer.


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Parabola General Equations - General Equations for JEE

An Introduction to Parabola

Last updated date: 19th Mar 2023
Total views: 72.6k
Views today: 0.51k

Parabola is a mathematical curve that is a part of the conic section. It is a plane curve which has a mirror symmetry and is approximately U shaped. The parabola fits several different descriptions in our everyday life and in even superficial situations both in real life and even in Physics. All these situations essentially define the same curve, that is parabola.

The parabola is one of the most important topics under the conic section. A section of a right circular cone cut by a plane which is parallel to the cone generator is basically known as a parabola. Its shape is traced by a quadratic equation and the graph of parabola shows the locus of a point that is always at a fixed distance from a stationary line. A point on the plane also remains fixed.

What Is Parabola?

The word "parabola" is made up of two words “para” and “bola”. “Para” means "for" and "bola" means "to throw". Therefore, the meaning is self-explanatory. "Parabola" refers to the shape or form of a curve created by kicking or tossing a ball into the air. A parabola can be defined as the locus of a point on a graph which moves in such a way that its distance from a particular fixed point that we call the focus is equal to its perpendicular distance from a fixed straight line called the directrix. The parabola is an important curve in coordinate geometry's conic sections.

Diagram of a Parabola

From the diagram above, we can see that S is the focus and ZM is the directrix. We have considered a point P on the parabola. So from our definition of the parabola we can write

PS=PM

Here PM is the perpendicular distance of the point from the directrix and PS is the distance of the point from the focus.

The Parabola in Real Life Scenario

Different Curves from a Conic Section

From the image above, we can visualise the parabola as a part of the conic section. Other conic sections include the circle, ellipse, and the hyperbola. All the conic sections have their own properties.

Important Definitions Related to the Parabola

Some important definitions have been described concerning the parabola, which are as follows:

S.No.

Terminology

Definition

The focus of the Parabola

A parabola is defined by a set of all points in a plane which are at equal distance from a given point and given line. The point is called the focus of the parabola.

Directrix of the Parabola

A parabola is defined by a set of all points in a plane which are at equal distance from a given point and given line. The line is called the directrix of the parabola.

Vertex

The vertex is the point at which the parabola and the axis of symmetry cross each other. The parabola curves dramatically at the vertex.

Focal Length

On the axis of symmetry, it is the distance between the vertex and the focus.

Latus Rectum

It's a line segment that runs parallel to the directrix and passes through the parabola's focus.

Axis of Symmetry

The line of symmetry of a parabola is a line that divides the parabola into two equal sections. It travels through the focus and is perpendicular to the directrix.

Eccentricity

Eccentricity is the ratio of a point's distance from the directrix to its distance from the focus. The parabola's eccentricity is 1.

The terms used in the above table have been depicted in the graph below.

Parts of a Parabola

General Equation of the Parabola

Here are the general equations of the parabola:

$y=a(x-h)^2+k$

$x=a(y-k)^2+h$

From the equations, we can deduce the nature of the parabola. The value of “a” in the equation of the parabola defines the direction of the parabola. The first equation is the equation of a regular parabola that opens towards the y axis. If the value of a is negative then the parabola opens towards the negative y axis and if it is positive then it opens towards the positive y axis.

The second equation is the equation of a sideways parabola that opens towards the x axis. Again, if a is positive then the parabola opens towards the positive x axis and if it is negative then it opens towards the negative x axis.

For a sideways parabola, we have some parabola formulas as listed below that help in deducing the important point of the parabola.

• (h,k) defines the vertex of the parabola.

• Latus rectum of the parabola is equal to 4a.

• The coordinates of the focus are  $\left(h,k+\dfrac{1}{4a}\right)$

• The directrix of the parabola is the line drawn perpendicular to the x-axis and passing the point (-a, 0). The directrix is perpendicular to the parabola's axis.

Derivation of the Parabola Equation

On the given parabola, there is a point P with the coordinates (x, y). The distance between point P and the directrix is equal to the distance between point P and the focus F, according to the definition of a parabola.

The eccentricity of the parabola is equal to 1. From the definition of parabola mentioned before, we can write the eccentricity e for the parabola as

$\dfrac{PB}{PF}=e$

$\dfrac{PB}{PF}=1$

$PB=PF$

Derivation of the Parabola Equation

The focus's coordinates are F(a,0). We can calculate its distance from P using the coordinate distance formula (x, y)

\begin{align} &P F=\sqrt{(x-a)^{2}+(y-0)^{2}} \\ &P F=\sqrt{(x-a)^{2}+(y)^{2}} \end{align}

The directrix's equation is x + a = 0, and we calculate PB using the perpendicular distance formula.

\begin{align} &P B=\dfrac{x+a}{\sqrt{1^{2}+0^{2}}} \\ &P B=x+a \\ &P B^{2}=\sqrt{(x+a)^{2}} \end{align}

We must use PF = PB to derive the parabola equation:

$\sqrt{(x-a)^{2}+y^{2}}=\sqrt{(x+a)^{2}}$

By squaring both sides of the equation,

\begin{align} &(x-a)^{2}+y^{2}=(x+a)^{2} \\ &x^{2}+a^{2}-2 a x+y^{2}=x^{2}+a^{2}+2 a x \\ &y^{2}=2 a x+2 a x \\ &y^{2}=4 a x \end{align}

We've now correctly deduced the typical parabola equation.

Now this was the equation of a parabola with its vertex at the origin. Suppose we have a parabola with its vertex at (h,k) then its equation is given as

$(y-k)^2=4a(x-h)$

We will see different types of parabolas in the next section. The equation of a parabola that we derived was opening towards the positive x-axis. There can be other parabolas opening towards the negative x-axis and the positive or negative y-axis.

Parabola Equation in Standard Form

Cartesian coordinates are a pair of numerical coordinates that represent points on a plane (x, y)

1. If P is Greater than Zero and P is on the Positive Y-Axis:

$x^{2}=4 p y$

Here, coordinates of F = (0, p) lie on the y-axis and p is the distance from vertex to focus on the y-axis.

Parabola x2 = 4 py with Coordinates of F = (0,P)

(b) If P is Less than Zero and P is on the Negative X-Axis:

$y^{2}=-4 p x$

Here, coordinates of F = (-p, 0) lie on the y-axis and p is the distance from vertex to focus on the x-axis.

Parabola with coordinates of F = (-P,0)

(c) If P is Less than Zero and P is on the Negative Y-Axis:

$x^{2}=-4 p y$

Here, coordinates of F = (0, -p) lie on the y-axis and p is the distance from vertex to focus on the y-axis.

Parabola with coordinates of F = (0,-P)

(d) If P is Greater than Zero and P is on the Positive X-Axis:

$y^{2}=4 p x$

Here, coordinates of F = (p, 0) lie on the y-axis and p is the distance from vertex to focus on the x-axis.

Parabola with coordinates of F = (P,0)

The observations from the above section and the diagrams of the parabola have been summarised and listed below.

• A parabola has axis-symmetric geometry. The axis of symmetry runs along the x-axis if the equation contains the term with y2, and along the y-axis if the term with x2 is present in the equation.

• The parabola opens to the right if the coefficient of x is positive and to the left if the coefficient of x is negative when the axis of symmetry is along the x-axis.

• The parabola widens upwards if the coefficient of y is positive and downwards if the coefficient of y is negative when the axis of symmetry is along the y-axis.

Parametric Equation of a Parabola

If we have a parabola with equation

y2 = 4ax , then the parametric equation of this parabola are

\begin{align} &x=a t^{2} \\ &y=2 a t \end{align}

Here t is the parameter of this parabola.

Any point on the parabola can be represented by the parametric equations as (at2, 2at)  and these points are called “t” points.

Diagram showing the t point of a parabola

From the above image, we can see the diagram of a parabola with its t point and the focus represented by (a,0).

Equation of a Chord Joining any Two Points of a Parabola

Suppose we have two points represented by the parameters t1 and t2 on a parabola. These points are given as (at12, 2at1)  and (at22, 2at2), then the equation of a chord joining these two points is given as,

\begin{align} &\left(y-2 a t_{1}\right)=\dfrac{2 a t_{2}-2 a t_{1}}{a t_{2}{ }^{2}-a t_{1}{ }^{2}}\left(x-a t_{1}{ }^{2}\right) \\ &y-2 a t_{1}=\dfrac{2 a\left(t_{2}-t_{1}\right)}{a\left(t_{1}+t_{2}\right)\left(t_{2}-t_{1}\right)}\left(x-a t_{1}{ }^{2}\right) \\ &y-2 a t_{1}=\dfrac{2}{t_{1}+t_{2}}\left(x-a t_{1}^{2}\right) \\ &\left(t_{1}+t_{2}\right) y=2 x+2 a t_{1} t_{2} \end{align}

A focal chord is a chord that passes through the focus of the parabola. For a focal chord,

t1t2=-1

Suppose now that one end of the focal chord is at (at2,2at) then the other end will be at $\left(\dfrac{a}{t^2},-2at\right)$

Position of a Point with Respect to a Parabola

Suppose we have a point (x1,y1). Now to know if the point lies inside, outside, or on the parabola, we have to check whether, (y12-4ax1) is greater than, less than, or equal to zero.

• If y12-4ax1>0, then the point lies outside the region of the parabola.

• If y12-4ax1<0, then the point lies inside the region of the parabola.

Similarly if y12-4ax1=0, then the point lies on the parabola.

Tangent to the Parabola

If a line represented by equation y=mx+c, touches a parabola y2=4ax then for the line to be a tangent to the parabola,

$c=\dfrac{a}{m}$

Similarly, if we have a parabola x2=4ay, then the line y=mx+c is a tangent to it if

$c=-am^2$

Now if the line is not given in the standard form then to check if the line is a tangent to a parabola, first, we need to eliminate one variable either x or y between the equations of the straight line and parabola and then apply the condition b2 = 4ac for the quadratic equation ax2+bx+c which is obtained.

The equation of a tangent to the parabola y2=4ax at a point (x1,y1) is given as

$yy_1=2a(x+x_1)$.

This equation is known as the point form equation of the tangent.

Similarly, the parametric form of a tangent to the parabola at a  point (at2,2at) is given as,

$ty=x+at^2$

The slope form of a tangent to a parabola y2=4ax is given as

$y=mx+\dfrac{a}{m}$

Here, m is the slope of the tangent.

Point of Intersection of Two Tangents

Suppose we have two tangents to a parabola y2=4ax at points t1 and t2The equation of the tangent at t1 will be

$yt_1=x+at_1^2$ ….(1)

Similarly the equation of the tangent at point t2 will be

$yt_2=x+at_2^2$.....(2)

Now, if we subtract equations (1) and (2), we get $y(t_1-t_2)=a(t_1^2-t_2^2)$

\begin{align} &y\left(t_{1}-t_{2}\right)=a\left(t_{1}^{2}-t_{2}^{2}\right) \\ &y=\dfrac{a\left(t_{1}+t_{2}\right)\left(t_{1}-t_{2}\right)}{\left(t_{1}-t_{2}\right)} \\ &y=a\left(t_{1}+t_{2}\right) \end{align}

Putting this value of y in equation (1), we get

\begin{align} &a\left(t_{1}+t_{2}\right) t_{1}=x+a t_{1}^{2} \\ &a t_{1}^{2}+a t_{1} t_{2}=x+a t_{1}^{2} \\ &x=a t_{1} t_{2} \end{align}

So the point of intersection of the tangents has coordinates

$x=at_1t_2$

$y=a(t_1+t_2)$

So the point of intersection is

$(a_1t_2, a(t_1+t_2))$

Point of intersection of 2 tangents

Normal to the Parabola

The tangent of a parabola is perpendicular to the parabola's normal.

Suppose we have a line y=mx+c and a parabola y2=4ax. This line is normal to the parabola when

$c=-2am-am^3$

Similarly, if the parabola is x2=4ay then the line y=mx+c is a normal to the parabola when

$c=2a+\dfrac{a}{m_2}$

Equation of Normal

The equation of a normal at point (x1,y1) to the parabola y2=4ax is given as

$y-y_1=-\dfrac{y_1}{2a}(x-x_1)$

This is the point form of the normal.

The parametric form of the equation of a normal at point (at2,2at1) is

$y+tx=2at+at^3$

Similarly the slope point form of the normal in terms of slope m is given as

$y=mx-2am-am^3$

Point of Intersection of Two Normals

Suppose we have two normals at points P(at12,2at1)  and Q(at22,2at2) on the parabola y2=4ax, then the point of intersection of the two normals represented by point R.

We know that normals and tangents are perpendicular to each other. As above mentioned that the slope of tangents is $\dfrac{1}{t_1}$ and $\dfrac{1}{t_2}$ so the slope of normals will be $-t_1$ and $-t_2$.

So writing equations of both normals;

$y-2a{t_1}=-{t_1}(x-a{t_1}^2)$ -----(1)

$y-2a{t_2}=-{t_2}(x-a{t_2}^2)$ -----(2)

Now we will solve equations (1) and (2) for finding the intersection point.

$2a{t_2}-{t_2}(x-a{t_2}^2)=2a{t_1}-{t_1}(x-a{t_1}^2)$

$x({t_2}-{t_1})=2a({t_2}-{t_1})+a({t_2}^3-{t_1}^3)$

$x=2a+a({t_2}^2+{t_1}^2+{t_1}{t_2})$

And substituting value of x in equation (1),

$y-2a{t_1}=-{t_1}(2a+a({t_2}^2+{t_1}^2+{t_1}{t_2})-a{t_1}^2)$

$y=2a{t_1}-{t_1}(2a+a{t_2}^2+a{t_1}{t_2})$

$y=-{t_1}(a{t_2}^2+a{t_1}{t_2})$

$y=-at_1t_2(t_1+t_2)$

The point of intersection of the normals is thus, $R(2a+a({t_2}^2+{t_1}^2+{t_1}{t_2}), -at_1t_2(t_1+t_2))$.

Conclusion

The parabolic graph is a symmetrical figure which is a 2D curve that represents the graph of a quadratic equation. Any point on the parabola maintains a fixed distance from the focus (point) and the directrix (straight line). The focal chord is any straight line passing through the focus of the parabola. The vertex of the parabola is the point of intersection between the parabolic graph and the line of symmetry. The slope of the normal of a parabola is the inverse of the slope of the tangent of the parabola. The general equation of a parabola is given by $y^{2}=4 a x$, where the vertex is at the origin (0, 0) and the axis is the y-axis.

Competitive Exams after 12th Science

FAQs on Parabola General Equations - General Equations for JEE

1. Where can parabolas be seen in real life?

There are a lot of real-life parabola examples that one encounters in everyday life. If you observe carefully, most bridges and flyovers are parabolic in nature. If you throw a ball upwards, the path taken by the ball before touching the ground is also a parabola. The roller coasters in the amusement parks are also classic real-life examples of the parabolic curve. Parabola has utilities in other fields as well. In economics, most of the profits, losses and forecasts are denoted by the parabolic graph. The height of the parabola represents the profits. The bigger the winnings, the higher the parabola.

2. What significance does the current topic have in terms of board exams? How much weight does it have in the JEE exam?

In the board exam, the topic parabola has intermediate levels. Since the board test is subjective in nature, the theoretical and numerical sections of this text are critical from an exam standpoint. Some numerical information is also available from the score advantage section. The weightage of the parabola is moderate because it only comprises a small amount of mathematics. The parabola is a simple idea that is used in this topic. Every year 2 to 3 questions from the parabola topic are asked in the JEE Main.

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# Math Help - circular permutation

1. ## circular permutation

Suppose there are $n$ families with $5$ people in each family for a total of $5n$ people arranged around a circular table. How many ways are there to seat the people such that each person sits next to another member of his family?

So fix $2$ people from each family (treat them as a unit). This leaves $(5n-2n)! = 3n!$ ways to arrange the rest?

2. You answer is definitely false.
just take n=1 as a counterexample for your solution.

3. Hint:for a family, there is only two way to be seated:
(1)all the members are seated togather as a unit.
(2)all the members are divided into two group consisting of 2 and 3 members, members in each group are seated togather as a unit, and two units are seperated by members of other family.

4. Originally Posted by Shanks
Hint:for a family, there is only two way to be seated:
(1)all the members are seated togather as a unit.
(2)all the members are divided into two group consisting of 2 and 3 members, members in each group are seated togather as a unit, and two units are seperated by members of other family.
So seat one family and there are $(n-1)!$ ways of arranging the other members.

5. No, It is possible that some fimalies are seperated into two Units(groups) by other family.
for example, here is a example of n=2 to clearify my posted thread:
(1)No family is seperated: that is all members in the same family are seated togather as a unit.
(2)For the case n=2, only one family are seperated is equivalent to (1). If
n>2, it is possible that one or some families are seperated by other family. We need a arguement on the number of families that are seperated.
(3)Two families are seperated, each family are divided into two groups consisting of 2 and 3 members.
Denote the two family members by $A_1,...,A_5; B_1,...,B_5$.
Here is a example that all members are seated in a circle clockwisely:
$A_1,A_2,B_1,B_2,A_3,A_4,A_5,B_3,B_4,B_5$
similarly, You can find all the ways that all members are seated in a circle such that each member sits next to a member of his family.


text:
# Why is a proxy needed in proxy re-encryption?

## Context

Here's the usual proxy re-encryption scenario:

• Alice, with key pair $$(pk_a, sk_a)$$ encrypts a message $$m$$, producing $$c = encrypt(pk_a, m)$$
• Alice saves $$c$$ on an untrusted storge provider (Dropbox, IPFS, etc..)
• Bob, with the key pair $$(pk_b, sk_b)$$ asks Alice to access $$c$$
• Alice generates $$re_{a\rightarrow b} = rekey(sk_a, pk_b)$$, which she sends to a proxy
• That proxy re-encrypts $$c$$ in $$c' = reencrypt(re_{a\rightarrow b}, c)$$
• Bob can now download $$c'$$ and obtain the plaintext $$m = decrypt(sk_b, c')$$

## Question

Why do we need a proxy? Can't Bob re-encrypt $$c$$ into $$c'$$ by himself? From what I've understood, the only role of the proxy is to re-encrypt $$c$$ and nothing more.

In other words: can Bob and the proxy be the same person/device?

## Further details

If Alice sends $$re_{a\rightarrow b}$$ to Bob via e-mail, or by letting it be publicly accessible (i.e. storing it on IPFS), Bob should be able to re-encrypt $$c$$ into $$c'$$ by himself, provided he has the right software to do so on his device.

What I'm also interested in is uderstanding whether there are any advantages for the re-encryption step being performed by a proxy and not by the recipient himslef. I can't think of a scenario where Bob may be forced to rely on a proxy.


text:
# The PyCASSO database: spatially resolved stellar population properties for CALIFA galaxies

DOI:
10.1093/mnras/stx1805
Publication date:
01/11/2017
Main author:
de Amorim, A. L.
IAA authors:
García-Benito, R.; Cortijo-Ferrero, C.; González Delgado, R. M.; López Fernández, R.; Pérez, E.
Authors:
de Amorim, A. L.; García-Benito, R.; Cid Fernandes, R.; Cortijo-Ferrero, C.; González Delgado, R. M.; Lacerda, E. A. D.; López Fernández, R.; Pérez, E.; Vale Asari, N.
Journal:
Monthly Notices of the Royal Astronomical Society
Refereed:
Yes
Publication type:
Article
Volume:
471
Pages:
3727-3752
Issue:
3
Abstract:

The Calar Alto Legacy Integral Field Area (CALIFA) survey, a pioneer in integral field spectroscopy legacy projects, has fostered many studies exploring the information encoded on the spatially resolved data on gaseous and stellar features in the optical range of galaxies. We describe a value-added catalogue of stellar population properties for CALIFA galaxies analysed with the spectral synthesis code STARLIGHT and processed with the PyCASSO platform. Our public data base (http://pycasso.ufsc.br/, mirror at http://pycasso.iaa.es/) comprises 445 galaxies from the CALIFA Data Release 3 with COMBO data. The catalogue provides maps for the stellar mass surface density, mean stellar ages and metallicities, stellar dust attenuation, star formation rates, and kinematics. Example applications both for individual galaxies and for statistical studies are presented to illustrate the power of this data set. We revisit and update a few of our own results on mass density radial profiles and on the local mass-metallicity relation. We also show how to employ the catalogue for new investigations, and show a pseudo Schmidt-Kennicutt relation entirely made with information extracted from the stellar continuum. Combinations to other databases are also illustrated. Among other results, we find a very good agreement between star formation rate surface densities derived from the stellar continuum and the Hα emission. This public catalogue joins the scientific community's effort towards transparency and reproducibility, and will be useful for researchers focusing on (or complementing their studies with) stellar properties of CALIFA galaxies.

Database:
WOK


text:
# 1118. Birds in Forest (25)-PAT甲级真题（并查集）

Some scientists took pictures of thousands of birds in a forest. Assume that all the birds appear in the same picture belong to the same tree. You are supposed to help the scientists to count the maximum number of trees in the forest, and for any pair of birds, tell if they are on the same tree.

#### Input Specification:

Each input file contains one test case. For each case, the first line contains a positive number N (<= 104) which is the number of pictures. Then N lines follow, each describes a picture in the format:
K B1 B2 … BK
where K is the number of birds in this picture, and Bi’s are the indices of birds. It is guaranteed that the birds in all the pictures are numbered continuously from 1 to some number that is no more than 104.

After the pictures there is a positive number Q (<= 104) which is the number of queries. Then Q lines follow, each contains the indices of two birds.

#### Output Specification:

For each test case, first output in a line the maximum possible number of trees and the number of birds. Then for each query, print in a line “Yes” if the two birds belong to the same tree, or “No” if not.

4
3 10 1 2
2 3 4
4 1 5 7 8
3 9 6 4
2
10 5
3 7

2 10
Yes
No

#### 分析

• 讲每幅画的小鸟进行Union操作，使其归属于同一个根
• 设置findFather函数用于查找小鸟所在集合的根，并把回溯过程中的节点的根都设置为对应的根节点
• 用exist数组记录遍历过的小鸟id，exist[id]=true
• cnt数组用于存储隶属同一个根节点的小鸟个数
• 遍历cnt数组并累加cnt数组不为0的个数，即为树的个数
• 判断两鸟是否在同一颗树上，只需要知道这两只鸟的根节点是否相等即可
#include
using namespace std;

int n, m, k;
const int maxn = 10010;
int fa[maxn] = {0}, cnt[maxn] = {0};

int findFather(int x) {
int a = x;
while (x != fa[x])
x = fa[x];   //一层层回溯
while (a != fa[a]) {
int z = a;
a = fa[a];
fa[z] = x;  //更新fa数组
}
return x;
}

void Union(int a, int b) {
int faA = findFather(a);
int faB = findFather(b);
if(faA!=faB)
fa[faA]=faB;
}

bool exist[maxn];

int main() {
cin >> n;
for (int i = 1; i <= maxn; i++)
fa[i] = i;
int id, tmp;
for (int i = 0; i < n; i++) {
cin >> k >> id;
exist[id]=true;
for (int j = 0; j < k - 1; j++) {
cin >> tmp;
Union(id, tmp);
exist[tmp] = true;
}
}

for (int i = 1; i <= maxn; i++) {
if (exist[i]) {
int root = findFather(i);
cnt[root]++;
}
}

int numTrees = 0, numBirds = 0;
for (int i = 1; i <= maxn; i++) {
if (exist[i] && cnt[i] != 0) {
numTrees++;
numBirds += cnt[i];
}
}
cout << numTrees << " " << numBirds << endl;
cin >> m;
int ida, idb;
for (int i = 0; i < m; i++) {
cin >> ida >> idb;
if (findFather(ida) == findFather(idb))
cout << "Yes" << endl;
else
cout << "No" << endl;
}
return 0;
}