Let the relation of knowledge to real life be very visible to your pupils and let them understand how by knowledge the world could be transformed. - BERTRAND RUSSELL

10.1 Introduction

In the preceding Chapter 10, we have studied various forms of the equations of a line. In this Chapter, we shall study about some other curves, viz., circles, ellipses, parabolas and hyperbolas. The names parabola and hyperbola are given by Apollonius. These curves are in fact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone. These curves have a very wide range of applications in fields such as planetary motion, design of telescopes and antennas, reflectors in flashlights

and automobile headlights, etc. Now, in the subsequent sections we will see how the intersection of a plane with a double napped right circular cone results in different types of curves.

10.2 Sections of a Cone

Let $l$ be a fixed vertical line and $m$ be another line intersecting it at a fixed point $V$ and inclined to it at angle $\alpha$ (Fig10.1).

Suppose we rotate the line $m$ around the line $l$ in such a way that the angle $\alpha$ remains constant. Then the surface generated is a double-napped right circular hollow cone herein after referred as cone and extending indefinitely far in both directions (Fig10.2).

The point $V$ is called the vertex; the line $l$ is the axis of the cone. The rotating line $m$ is called a generator of the cone. The vertex separates the cone into two parts called nappes.

If we take the intersection of a plane with a cone, the section so obtained is called a conic section. Thus, conic sections are the curves obtained by intersecting a right circular cone by a plane.

We obtain different kinds of conic sections depending on the position of the intersecting plane with respect to the cone and by the angle made by it with the vertical axis of the cone. Let $\beta$ be the angle made by the intersecting plane with the vertical axis of the cone (Fig10.3).

The intersection of the plane with the cone can take place either at the vertex of the cone or at any other part of the nappe either below or above the vertex.

10.2.1 Circle, ellipse, parabola and hyperbola

When the plane cuts the nappe (other than the vertex) of the cone, we have the following situations:

(a) When $\beta=90^{\circ}$, the section is a circle (Fig10.4).

(b) When $\alpha<\beta<90^{\circ}$, the section is an ellipse (Fig10.5).

(c) When $\beta=\alpha$; the section is a parabola (Fig10.6).

(In each of the above three situations, the plane cuts entirely across one nappe of the cone).

(d) When $0 \leq \beta<\alpha$; the plane cuts through both the nappes and the curves of intersection is a hyperbola (Fig10.7).

10.2.2 Degenerated conic sections

When the plane cuts at the vertex of the cone, we have the following different cases:

(a) When $\alpha<\beta \leq 90^{\circ}$, then the section is a point (Fig10.8).

(b) When $\beta=\alpha$, the plane contains a generator of the cone and the section is a straight line (Fig10.9).

It is the degenerated case of a parabola.

(c) When $0 \leq \beta<\alpha$, the section is a pair of intersecting straight lines (Fig10.10). It is the degenerated case of a hyperbola.

In the following sections, we shall obtain the equations of each of these conic sections in standard form by defining them based on geometric properties.

10.3 Circle

Definition 1 A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.

The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle (Fig 10.11).

The equation of the circle is simplest if the centre of the circle is at the origin. However, we derive below the equation of the circle with a given centre and radius (Fig 10.12).

Given $C(h, k)$ be the centre and $r$ the radius of circle. Let $P(x, y)$ be any point on the circle (Fig10.12). Then, by the definition, $|CP|=r$. By the distance formula, we have

i.e.

$ \begin{aligned} & \sqrt{(x-h)^{2}+-k)^{2}}=r & (x-h)^{2}+(y-k)^{2}=r^{2} \end{aligned} $

This is the required equation of the circle with centre at $(h, k)$ and radius $r$.

10.4 Parabola

Definition 2 A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane.

The fixed line is called the directrix of the parabola and the fixed point $F$ is called the focus (Fig 10.13). (‘Para’ means ‘for’ and ‘bola’ means ’throwing’, i.e., the shape described when you throw a ball in the air).

Note - If the fixed point lies on the fixed line, then the set of points in the plane, which are equidistant from the fixed point and the fixed line is the straight line through the fixed point and perpendicular to the fixed line. We call this straight line as degenerate case of the parabola.

A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola (Fig10.14).

10.4.1 Standard equations of parabola

The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the $x$-axis or $y$-axis. The four possible such orientations of parabola are shown below in Fig10.15 (a) to (d).

We will derive the equation for the parabola shown above in Fig 10.15 (a) with focus at $(a, 0) a>0$; and directricx $x=-a$ as below:

Let $F$ be the focus and $l$ the directrix. Let FM be perpendicular to the directrix and bisect FM at the point O. Produce MO to X. By the $(-a, y)$ B definition of parabola, the mid-point $O$ is on the parabola and is called the vertex of the parabola. Take $O$ as origin, $OX$ the $x$-axis and $OY$ perpendicular to it as the $y$-axis. Let the distance from the directrix to the focus be $2 a$. Then, the coordinates of the focus are $(a, 0)$, and the equation of the directrix is $x+a=0$ as in Fig10.16.

Let $P(x, y)$ be any point on the parabola such that

$ PF=PB, \quad \quad\quad\quad\quad\quad\quad\quad\quad\ldots(1) $

where $PB$ is perpendicular to $l$. The coordinates of $B$ are $(-a, y)$. By the distance formula, we have

$ PF=\sqrt{(x-a)^{2}+y^{2}} \text{ and } PB=\sqrt{(x+a)^{2}} $

Since $PF=PB$, we have

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

i.e. $ \quad\quad\quad(x-a)^{2}+y^{2}=(x+a)^{2}$

or $\quad\quad\quad x^{2}-2 a x+a^{2}+y^{2}=x^{2}+2 a x+a^{2}$

or $\quad\quad\quad y^{2}=4 a x(a>0)$.

Hence, any point on the parabola satisfies

$ y^{2}=4 a x \quad \quad\quad\quad\quad\quad\quad\quad\quad\ldots(2) $

Conversely, let $P(x, y)$ satisfy the equation (2)

$ \begin{aligned} PF & =\sqrt{(x-a)^{2}+y^{2}} \quad=\sqrt{(x-a)^{2}+4 a x} \\ & =\sqrt{(x+a)^{2}}=PB \quad \quad\quad\quad\quad\quad\quad\quad\quad\ldots(3) \end{aligned} $

and so $P(x, y)$ lies on the parabola.

Thus, from (2) and (3) we have proved that the equation to the parabola with vertex at the origin, focus at $(a, 0)$ and directrix $x=-a$ is $y^{2}=4 a x$.

Discussion In equation (2), since $a>0, x$ can assume any positive value or zero but no negative value and the curve extends indefinitely far into the first and the fourth quadrants. The axis of the parabola is the positive $x$-axis.

Similarly, we can derive the equations of the parabolas in:

Fig 11.15 (b) as $y^{2}=-4 a x$,

Fig 11.15 (c) as $x^{2}=4 a y$,

Fig $11.15(d)$ as $x^{2}=-4 a y$,

These four equations are known as standard equations of parabolas.

Note - The standard equations of parabolas have focus on one of the coordinate axis; vertex at the origin and thereby the directrix is parallel to the other coordinate axis. However, the study of the equations of parabolas with focus at any point and any line as directrix is beyond the scope here.

From the standard equations of the parabolas, Fig10.15, we have the following observations:

1. Parabola is symmetric with respect to the axis of the parabola.If the equation has a $y^{2}$ term, then the axis of symmetry is along the $x$-axis and if the equation has an $x^{2}$ term, then the axis of symmetry is along the $y$-axis.

2. When the axis of symmetry is along the $x$-axis the parabola opens to the

(a) right if the coefficient of $x$ is positive,

(b) left if the coefficient of $x$ is negative.

3. When the axis of symmetry is along the $y$-axis the parabola opens

(c) upwards if the coefficient of $y$ is positive.

(d) downwards if the coefficient of $y$ is negative.

10.4.2 Latus rectum

Definition 3 Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola (Fig10.17).

To find the Length of the latus rectum of the parabola $ y^{2}= 4 a x $ (Fig 10.18).

By the definition of the parabola, $AF=AC$.

But $ \quad \quad \quad AC=FM=2 a $

Hence $ \quad \quad \quad AF=2 a $

And since the parabola is symmetric with respect to $x$-axis $AF=FB$ and so

$AB=$ Length of the latus rectum $=4 a$.

10.5 Ellipse

Definition 4 An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.

The two fixed points are called the foci (plural of ‘focus’) of the ellipse (Fig10.20).

Note - The constant which is the sum of the distances of a point on the ellipse from the two fixed points is always greater than the distance between the two fixed points is always greater than the distance between the two fixed points.

The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse(Fig 10.21).

We denote the length of the major axis by $2 a$, the length of the minor axis by $2 b$ and the distance between the foci by $2 c$. Thus, the length of the semi major axis is $a$ and semi-minor axis is $b$ (Fig10.22).

10.5.1 Relationship between semi-major axis, semi-minor axis and the distance of the focus from the center fo the ellipse (Fig 10.23)

Take a point $P$ at one end of the minor axis.

Sum of the distances from the point $P$ to the foci is $F_1P + F_2P = F_1O + OP + F_2P$

(Since, $F_1P = F_1O + OP$)

$\quad \quad \quad \quad \quad = c + a +a - c = 2a$

Take a point Q at one end of the minor axis.

Sum of the distances from the point Q to the foci is

$ F_1 P+F_2 Q=\sqrt{b^{2}+c^{2}}+\sqrt{b^{2}+c^{2}}=2 \sqrt{b^{2}+c^{2}} $

Since both $P$ and $Q$ lies on the ellipse.

By the definition of ellipse, we have

$ \begin{aligned} 2 \sqrt{b^{2}+c^{2}} & =2 a, \text{ i.e., } \quad a=\sqrt{b^{2}+c^{2}} \\ \text{or} \quad \quad \quad \quad a^{2} & =b^{2}+c^{2}, \text{ i.e., } c=\sqrt{a^{2}-b^{2}} \end{aligned} $

10.5.2 Eccentricity

Definition 5 The eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse (eccentricity is denoted by $e$ ) i.e., $e=\frac{c}{a}$.

Then since the focus is at a distance of $c$ from the centre, in terms of the eccentricity the focus is at a distance of ae from the centre.

10.5.3 Standard equations of an ellipse

The equation of an ellipse is simplest if the centre of the ellipse is at the origin and the foci are on the $x$-axis or $y$-axis. The two such possible orientations are shown in Fig 10.24.

We will derive the equation for the ellipse shown above in Fig 10.24 (a) with foci on the $x$-axis.

Let $F_1$ and $F_2$ be the foci and $O$ be the mid-point of the line segment $F_1 F_2$. Let $O$ be the origin and the line from $O$ through $F_2$ be the positive $x$-axis and that through $F_1$ as the negative $x$-axis. Let, the line through $O$ perpendicular to the $x$-axis be the $y$-axis. Let the coordinates of $F_1$ be $(-c, 0)$ and $F_2$ be $(c, 0)$ (Fig 10.25).

Let $P(x, y)$ be any point on the ellipse such that the sum of the distances from $P$ to the two foci be $2 a$ so given

$ PF_1+PF_2=2 a . \quad \quad \quad \quad \quad \quad \quad \ldots (1) $

Using the distance formula, we have

$ \begin{aligned} & \qquad \sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}=2 a \\ & \text{ i.e., } \sqrt{(x+c)^{2}+y^{2}}=2 a-\sqrt{(x-c)^{2}+y^{2}} \end{aligned} $

$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $

Squaring both sides, we get

$ (x+c)^{2}+y^{2}=4 a^{2}-4 a \sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2} $

which on simplification gives

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

Squaring again and simplifying, we get

$ \begin{aligned} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}-c^{2}}=1 & \\ \text{ i.e., } \quad \quad\quad \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 & (\text{ Since } c^{2}=a^{2}-b^{2}) \end{aligned} $

Hence any point on the ellipse satisfies

$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $

Conversely, let $P(x, y)$ satisfy the equation (2) with $0<c<a$. Then

$ y^{2}=b^{2}(1-\frac{x^{2}}{a^{2}}) $

Therefore, $PF_1=\sqrt{(x+c)^{2}+y^{2}}$

$ \begin{aligned} & =\sqrt{(x+c)^{2}+b^{2}(\frac{a^{2}-x^{2}}{a^{2}})} \\ & =\sqrt{(x+c)^{2}+(a^{2}-c^{2})(\frac{a^{2}-x^{2}}{a^{2}})}(\text{ since } b^{2}=a^{2}-c^{2}) \\ & =\sqrt{(a+\frac{c x}{a})^{2}}=a+\frac{c}{a} x \end{aligned} $

Similarly $\quad PF_2=a-\frac{c}{a} x$

Hence $\quad PF_1+PF_2=a+\frac{c}{a} x+a-\frac{c}{a} x=2 a \quad \quad \quad \quad \quad \ldots(3)$

So, any point that satisfies $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, satisfies the geometric condition and so $P(x, y)$ lies on the ellipse.

Hence from (2) and (3), we proved that the equation of an ellipse with centre of the origin and major axis along the $x$-axis is

$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $

Discussion From the equation of the ellipse obtained above, it follows that for every point $P(x, y)$ on the ellipse, we have

$ \frac{x^{2}}{a^{2}}=1-\frac{y^{2}}{b^{2}} \leq 1 \text{, i.e., } x^{2} \leq a^{2} \text{, so }-a \leq x \leq a \text{. } $

Therefore, the ellipse lies between the lines $x=-a$ and $x=a$ and touches these lines.

Similarly, the ellipse lies between the lines $y=-b$ and $y=b$ and touches these lines.

Similarly, we can derive the equation of the ellipse in Fig 10.24 (b) as $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$. These two equations are known as standard equations of the ellipses.

Note - The standard equations of ellipses have centre at the origin and the major and minor axis are coordinate axes. However, the study of the ellipses with centre at any other point, and any line through the centre as major and the minor axes passing through the centre and perpendicular to major axis are beyond the scope here.

From the standard equations of the ellipses (Fig10.24), we have the following observations:

1. Ellipse is symmetric with respect to both the coordinate axes since if $(x, y)$ is a point on the ellipse, then $(-x, y),(x,-y)$ and $(-x,-y)$ are also points on the ellipse.

2. The foci always lie on the major axis. The major axis can be determined by finding the intercepts on the axes of symmetry. That is, major axis is along the $x$-axis if the coefficient of $x^{2}$ has the larger denominator and it is along the $y$-axis if the coefficient of $y^{2}$ has the larger denominator.

10.5.4 Latus rectum

Definition 6 Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse (Fig 10.26).

To find the length of the latus rectum of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Let the length of $AF_2$ be $l$.

Then the coordinates of $A$ are $(c, l)$,i.e., $(a e, l)$

Since A lies on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, we have

$ \begin{aligned} & \frac{(a e)^{2}}{a^{2}}+\frac{l^{2}}{b^{2}}=1 \\ & \Rightarrow l^{2}=b^{2}(1-e^{2}) \\ & \text{But} \quad \quad \quad e^{2}=\frac{c^{2}}{a^{2}}=\frac{a^{2}-b^{2}}{a^{2}}=1-\frac{b^{2}}{a^{2}} \end{aligned} $

Therefore $ \quad \quad \quad l^{2}=\frac{b^{4}}{a^{2}}, \text{ i.e., } l=\frac{b^{2}}{a} $

Since the ellipse is symmetric with respect to $y$-axis (of course, it is symmetric w.r.t. both the coordinate axes), $AF_2=F_2 B$ and so length of the latus rectum is $\frac{2 b^{2}}{a}$.

10.6 Hyperbola

Definition 7 A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

The term “difference” that is used in the definition means the distance to the farther point minus the distance to the closer point. The two fixed points are called the foci of the hyperbola. The mid-point of the line segment joining the foci is called the centre of the hyperbola. The line through the foci is called the transverse axis and the line through the centre and perpendicular to the transverse axis is called the conjugate axis. The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola (Fig 10.27).

We denote the distance between the two foci by $2 c$, the distance between two vertices (the length of the transverse axis) by $2 a$ and we define the quantity $b$ as

$b=\sqrt{c^{2}-a^{2}}$

Also $2 b$ is the length of the conjugate axis (Fig 10.28).

To find the constant $P_1 F_2-P_1 F_1$ :

By taking the point $P$ at $A$ and $B$ in the Fig 10.28, we have

$BF_1-BF_2=AF_2-AF_1$ (by the definition of the hyperbola)

$BA+AF_1-BF_2=AB+BF_2-AF_1$

i.e., $AF_1=BF_2$

So that, $BF_1-BF_2=BA+AF_1-BF_2=BA=2 a$

10.6.1 Eccentricity

Definition 8 Just like an ellipse, the ratio $e=\frac{c}{a}$ is called the eccentricity of the hyperbola. Since $c \geq a$, the eccentricity is never less than one. In terms of the eccentricity, the foci are at a distance of ae from the centre.

10.6.2 Standard equation of Hyperbola

The equation of a hyperbola is simplest if the centre of the hyperbola is at the origin and the foci are on the $x$-axis or $y$-axis. The two such possible orientations are shown in Fig10.29.

We will derive the equation for the hyperbola shown in Fig 10.29(a) with foci on the $x$-axis.

Let $F_1$ and $F_2$ be the foci and $O$ be the mid-point of the line segment $F_1 F_2$. Let $O$ be the origin and the line through $O$ through $F_2$ be the positive $x$-axis and that through $F_1$ as the negative $x$-axis. The line through $O$ perpendicular to the $x$-axis be the $y$-axis. Let the coordinates of $F_1$ be $\mathbf X^{\prime}$ $(-c, 0)$ and $F_2$ be $(c, 0)$ (Fig 10.30).

Let $P(x, y)$ be any point on the hyperbola such that the difference of the distances from $P$ to the farther point minus the closer point be $2 a$. So given, $PF_1-PF_2=2 a$

Using the distance formula, we have

$ \begin{aligned} & \quad \quad \quad \quad \sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=2 a \\ &\text{i.e., } \quad \quad \quad \sqrt{(x+c)^{2}+y^{2}}=2 a+\sqrt{(x-c)^{2}+y^{2}} \end{aligned} $

Squaring both side, we get

$ (x+c)^{2}+y^{2}=4 a^{2}+4 a \quad \sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2} $

and on simplifying, we get

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

On squaring again and further simplifying, we get

$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{c^{2}-a^{2}}=1 $

i.e., $\quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad$ (Since $c^{2}-a^{2}=b^{2}$ )

Hence any point on the hyperbola satisfies $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

Conversely, let $P(x, y)$ satisfy the above equation with $0<a<c$. Then

$ y^{2}=b^{2}(\frac{x^{2}-a^{2}}{a^{2}}) $

Therefore,

$ \begin{aligned} PF_1 & =+\sqrt{(x+c)^{2}+y^{2}} \\ & =+\sqrt{(x+c)^{2}+b^{2}(\frac{x^{2}-a^{2}}{a^{2}})}=a+\frac{c}{a} x \end{aligned} $

Similarly, $\quad PF_2=a-\frac{a}{c} x$

In hyperbola $c>a$; and since $P$ is to the right of the line $x=a, x>a, \frac{c}{a} x>a$. Therefore,

$ a-\frac{c}{a} x \text{ becomes negative. Thus, } PF_2=\frac{c}{a} x-a \text{. } $

Therefore $ \quad \quad \quad \quad PF_1-PF_2=a+\frac{c}{a} x-\frac{c x}{a}+a=2 a $

Also, note that if $P$ is to the left of the line $x=-a$, then

$ PF_1=-(a+\frac{c}{a} x), PF_2=a-\frac{c}{a} x $

In that case $PF_2-PF_1=2 a$. So, any point that satisfies $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, lies on the hyperbola.

Thus, we proved that the equation of hyperbola with origin $(0,0)$ and transverse axis along $x$-axis is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$.

Note - A hyperbola in which $a=b$ is called an equilateral hyperbola.

Discussion From the equation of the hyperbola we have obtained, it follows that, we have for every point $(x, y)$ on the hyperbola, $\frac{x^{2}}{a^{2}}=1+\frac{y^{2}}{b^{2}} \geq 1$.

i.e, $|\frac{x}{a}| \geq 1$, i.e., $x \leq-a$ or $x \geq a$. Therefore, no portion of the curve lies between the lines $x=+a$ and $x=-a$, (i.e. no real intercept on the conjugate axis).

Similarly, we can derive the equation of the hyperbola in Fig 11.31 (b) as $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ These two equations are known as the standard equations of hyperbolas.

Note - The standard equations of hyperbolas have transverse and conjugate axes as the coordinate axes and the centre at the origin. However, there are hyperbolas with any two perpendicular lines as transverse and conjugate axes, but the study of such cases will be dealt in higher classes.

From the standard equations of hyperbolas (Fig10.27), we have the following observations:

1. Hyperbola is symmetric with respect to both the axes, since if $(x, y)$ is a point on the hyperbola, then $(-x, y),(x,-y)$ and $(-x,-y)$ are also points on the hyperbola.

2. The foci are always on the transverse axis. It is the positive term whose denominator gives the transverse axis. For example, $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$

has transverse axis along $x$-axis of length 6 , while $\frac{y^{2}}{25}-\frac{x^{2}}{16}=1$ has transverse axis along y-axis of length 10 .

10.6.3 Latus rectum

Definition 9 Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.

As in ellipse, it is easy to show that the length of the latus rectum in hyperbola is $\frac{2 b^{2}}{a}$.

Summary

In this Chapter the following concepts and generalisations are studied.

A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.

The equation of a circle with centre $(h, k)$ and the radius $r$ is

$ (x-h)^{2}+(y-k)^{2}=r^{2} . $

A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.

The equation of the parabola with focus at $(a, 0) a>0$ and directrix $x=-a$ is

$ y^{2}=4 a x $

Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.

Length of the latus rectum of the parabola $y^{2}=4 a x$ is $4 a$.

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.

The equation of an ellipse with foci on the $x$-axis is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$.

Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse.

Length of the latus rectum of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is $\frac{2 b^{2}}{a}$.

The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

The equation of a hyperbola with foci on the $x$-axis is : $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.

Length of the latus rectum of the hyperbola $: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is : $\frac{2 b^{2}}{a}$.

The eccentricity of a hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola.

Historical Note

Geometry is one of the most ancient branches of mathematics. The Greek geometers investigated the properties of many curves that have theoretical and practical importance. Euclid wrote his treatise on geometry around 300 B.C. He was the first who organised the geometric figures based on certain axioms suggested by physical considerations. Geometry as initially studied by the ancient Indians and Greeks, who made essentially no use of the process of algebra. The synthetic approach to the subject of geometry as given by Euclid and in Sulbasutras, etc., was continued for some 1300 years. In the 200 B.C., Apollonius wrote a book called ‘The Conic’ which was all about conic sections with many important discoveries that have remained unsurpassed for eighteen centuries.

Modern analytic geometry is called ‘Cartesian’ after the name of Rene Descartes (1596-1650) whose relevant ‘La Geometrie’ was published in 1637. But the fundamental principle and method of analytical geometry were already discovered by Pierre de Fermat (1601-1665). Unfortunately, Fermats treatise on the subject, entitled Ad Locus Planos et So LIDOS Isagoge (Introduction to Plane and Solid Loci) was published only posthumously in 1679. So, Descartes came to be regarded as the unique inventor of the analytical geometry.

Isaac Barrow avoided using cartesian method. Newton used method of undetermined coefficients to find equations of curves. He used several types of coordinates including polar and bipolar. Leibnitz used the terms ‘abscissa’, ‘ordinate’ and ‘coordinate’. L’ Hospital (about 1700) wrote an important textbook on analytical geometry.

Clairaut (1729) was the first to give the distance formula although in clumsy form. He also gave the intercept form of the linear equation. Cramer (1750) made formal use of the two axes and gave the equation of a circle as

$ (y-a)^{2}+(b-x)^{2}=r $

He gave the best exposition of the analytical geometry of his time. Monge (1781) gave the modern ‘point-slope’ form of equation of a line as

$ y-y^{\prime}=a(x-x^{\prime}) $

and the condition of perpendicularity of two lines as $a a^{\prime}+1=0$.

S.F. Lacroix (1765-1843) was a prolific textbook writer, but his contributions to analytical geometry are found scattered. He gave the ’two-point’ form of equation of a line as

$ y-\beta=\frac{\beta^{\prime}-\beta}{\alpha^{\prime}-\alpha}(x-\alpha) $

and the length of the perpendicular from $(\alpha, \beta)$ on $y=a x+b$ as $\frac{(\beta-a-b)}{\sqrt{1+a^{2}}}$.

His formula for finding angle between two lines was $\tan \theta=(\frac{a^{\prime}-a}{1+a a^{\prime}})$. It is, of course, surprising that one has to wait for more than 150 years after the invention of analytical geometry before finding such essential basic formula. In 1818, C. Lame, a civil engineer, gave $m E+m^{\prime} E^{\prime}=0$ as the curve passing through the points of intersection of two loci $E=0$ and $E^{\prime}=0$.

Many important discoveries, both in Mathematics and Science, have been linked to the conic sections. The Greeks particularly Archimedes (287-212 B.C.) and Apollonius (200 B.C.) studied conic sections for their own beauty. These curves are important tools for present day exploration of outer space and also for research into behaviour of atomic particles.