Topics covered

1. Auxilliary circle

2. Eccentric angle

3. Equation of chord

4. Position of a point with respect to an ellipse.

1. Auxiliary Circle

The circle described on the major axis of an ellipse as diameter is called an auxiliary circle of the ellipse

If $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is an ellipse then its auxiliary circle is $x^2+y^2=a^2$

2. Eccentric angle of a point

Let $\mathrm{P}$ be any point on the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$

Draw PM perpendicular to major axis from $\mathrm{P}$ and produce MP to meet the auxiliary circle at Q. Join CQ. $\mathrm{\angle }\mathrm{QCA}=\theta$ is called eccentric angle of point $\mathrm{P}$ Note that the angle ACP is not eccentric angle. i.e. eccentric angle of $\mathrm{P}$ on an ellipse is the angle which the radius through the corresponding point on, the auxiliary circle makes with the major axis

$\therefore \mathrm{Q}(\mathrm{acos}\theta ,\mathrm{asin}\theta )$

$\therefore \mathrm{x}$-coordinate of $\mathrm{P}$ is acos $\theta$

$\frac{{\mathrm{a}}^2{\cos }^2\theta }{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$

$\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1-{\cos }^2\theta$

${\mathrm{y}}^2={\mathrm{b}}^2{\sin }^2\theta$

$\mathrm{y}=\mathrm{b}\sin \theta$

$\therefore$ Coordinate of $\mathrm{P}$ is $(\mathrm{acos}\theta ,\mathrm{bsin}\theta )$

i.e. $x=a\cos \theta$ and $y=b\sin \theta$ is the parameter equations of the ellipse.

$(a\cos \theta ,b\sin \theta )$ is also called the point ’ $\theta$ '

3. Equation of the chord

$\mathrm{Let}\mathrm{P}(\mathrm{acos}\theta ,b\sin \theta )$ and $\mathrm{Q}(\mathrm{acos}\varphi ,\mathrm{bsin}\varphi )$ be any two points of the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ then the equation of the chord joining these two points is

$y-b\sin \theta =\frac{b\sin \varphi -b\sin \theta }{a\cos \varphi -a\cos \theta }(x-a\cos \theta )$

Simplifying the equation we get

$\frac{\mathrm{x}}{\mathrm{a}}\cos \left(\frac{\theta +\varphi }{2}\right)+\frac{\mathrm{y}}{\mathrm{b}}\sin \left(\frac{\theta +\varphi }{2}\right)=\cos \frac{\theta -\varphi }{2}$

$\theta \mathrm{\&}\varphi$ are eccentric angle of points $\mathrm{P}$ and $\mathrm{Q}$ of ellipse

4. Position of a point $(\mathrm{h},\mathrm{k})$ with respect to an ellipse

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

Now P will lie outside, on or inside the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ according as

$\frac{{\mathrm{h}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{k}}^2}{{\mathrm{b}}^2}-1>,=,<0$

Examples

1. Find the equation of the curve whose parametric equation are $x=1+4\cos \theta ,y=2+3\sin \theta \in \mathrm{R}$

2. Find the eccentric angle of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance from the centre of the ellipse is $\sqrt{5}$

3. If $\alpha$ and $\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is

(a) $\frac{\cos \alpha +\cos \beta }{\cos (\alpha -\beta )}$

(b) $\frac{\sin \alpha -\sin \beta }{\sin (\alpha -\beta )}$

(c) $\frac{\cos \alpha -\cos \beta }{\cos (\alpha -\beta )}$

(d) $\frac{\sin \alpha +\sin \beta }{\sin (\alpha +\beta )}$

4. An ellipse passes through the point $(4,-1)$ and touches the line $x+4y-10=0$. Find its equation of its axes coincide with coordinate axes.

5. If $\frac{x}{a}+\frac{y}{b}=\sqrt{2}$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then find its eccentric angle $\theta$ of point of contact.

Practice questions

1. The sum of the squares of the reciprocals of two perpendicular diameter of an ellipse is

(a) $\frac{1}{4}(\frac{1}{{\mathrm{a}}^2}+\frac{1}{{\mathrm{b}}^2})$

(b) $\frac{1}{2}(\frac{1}{{\mathrm{a}}^2}+\frac{1}{{\mathrm{b}}^2})$

(c) $\frac{1}{{\mathrm{a}}^2}+\frac{1}{{\mathrm{b}}^2}$

(d) None of these

2. Prove that any point on the ellipse whose foci are $(-1,0)$ and $(7,0)$ and eccentcicity $\frac{1}{2}$ is $(3+8\cos \theta ,4\sqrt{3}\sin \theta ),\theta \in R$. Also find the eq of the ellipse

3. Let $\mathrm{E}$ be the ellipse $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$ and $\mathrm{C}$ be the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=9$. Let $\mathrm{P}$ and $\mathrm{Q}$ be the points

$(1,2)$ and $(2,1)$ respectively. Then

(a) $Q$ lies inside $\mathrm{C}$ but outside $\mathrm{E}$

(b) $\mathrm{Q}$ lies outside both $\mathrm{C}$ and $\mathrm{E}$

(c) $\mathrm{P}$ lies inside both $\mathrm{C}$ and $\mathrm{E}$

(d) $\mathrm{P}$ lies inside $\mathrm{C}$ but outside $\mathrm{E}$

4. $\mathrm{P}$ is a variable on the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ with ${\mathrm{AA}}^{\prime }$ as the major axis. Then the maximum area of the triangle ${\mathrm{APA}}^{\prime }$ is

(a) $\mathrm{ab}$

(b) $2\mathrm{ab}$

(c) $\mathrm{ab}/2$

(d) None of these

5. A man running round a race course notes that the sum of the distances of two flag-posts from him is always $10\mathrm{m}$ and the distance between the flag-posts is $8\mathrm{m}$. The area of the path he encloses in square meters is

(a) $15\pi$

(b) $12\pi$

(c) $18\pi$

(d) $8\pi$

6. If the line $\ell \mathrm{x}+\mathrm{my}+\mathrm{n}=0$ cuts the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{25}=1$ in points whose eccentric angles differ by $\frac{\pi }{2}$ then $\frac{{\mathrm{a}}^2{\ell }^2+{\mathrm{b}}^2{\mathrm{m}}^2}{{\mathrm{n}}^2}$

(a) $1$

(b) $2$

(c) $4$

(d) $3/2$

7. If $\mathrm{PSQ}$ is a focal chord if the ellipse $16{\mathrm{x}}^2+25{\mathrm{y}}^2=400$ such that $\mathrm{SP}=8$, then $\mathrm{SQ}=$

(a) 1

(b) 2

(c) 3

(d) 4

8. If equation of the ellipse is $2x^2+3y^2-8x+6y+5=0$ then which of the following are true?

(a) equation of director circle is $x^2+y^2-4x+2y=10$

(b) director circle will pass through $(4,-1)$

(c) equation of auxillary circle is $x^2+y^2-4x+2y+2=0$

(d) None of these

9. The foci of ellipse ${\left(\frac{x}{5}\right)}^2+{\left(\frac{y}{3}\right)}^2=1$ are $S$ and $S^{\prime }$. P is a point on ellipse whose eccentric angle is $\pi /3$. The incentre of triangle ${\mathrm{SPS}}^{\prime }$ is

(a) $(2,\sqrt{3})$

(b) $(2,\frac{2}{\sqrt{3}})$

(c) $(2,\frac{\sqrt{3}}{2})$

(d) $(\sqrt{3},2)$