Name: Parametric Equation of an Ellipse
Uploaded: 2019-07-19
Duration: 7 min 12 s
Description: In this video, you will understand the parametric equation of an ellipse with examples, which will help you solve questions in the exam. Watch and learn

Question

A chord of ellipse

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

whose eccentric angles of extremities are

α

and

β

, intersects its director circle at point

A

and

B

. Tangents at

A

and

B

intersect at point

P

. Find the equation of circumcircle of triangle

A B P

.

Accepted Answer

We have,

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 . . . . (i)$

Equation of director circle of an ellipse is

$A B P$

Equation of chord of contact $A B$ w.r.t. circle $(ii)$ is

$x h + y k = a^{2} + b^{2} . . . . (iii)$

Also, $A B$ is chord to en ellipse, so equation of $A B$ can be written as

$\frac{x}{a} \cos (\frac{α + β}{2}) + \frac{y}{b} \sin (\frac{α + β}{2}) = 1 . . . . (iv)$

Equation $(iii) & (iv)$ are representing same line, hence

$\frac{h}{\frac{1}{a} \cos (\frac{α + β}{2})} = \frac{k}{\frac{1}{b} \sin (\frac{α + β}{2})} = \frac{a^{2} + b^{2}}{1}$

Therefore,

$h = (\frac{a^{2} + b^{2}}{a}) \cos (\frac{α + β}{2})$ and $k = (\frac{a^{2} + b^{2}}{b}) \sin (\frac{α + β}{2})$

Now, $P, A, O, B$ are concyclic, hence, circumcircle of $∆ P A B$ is passing through the centre of ellipse and director circle, i.e., origin.

Hence, $O P$ is the diameter of the required circumcircle. Its, equation is

$(x - h) (x - 0) + (y - 0) (y - k) = 0$

$\Rightarrow x (x - h) + y (y - k) = 0$

$\Rightarrow x (x - (\frac{a^{2} + b^{2}}{a}) \cos (\frac{α + β}{2})) + y (y - (\frac{a^{2} + b^{2}}{b}) \sin (\frac{α + β}{2})) = 0$

A chord of ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ whose eccentric angles of extremities are $α$ and $β$ , intersects its director circle at point $A$ and $B$ . Tangents at $A$ and $B$ intersect at point $P$ . Find the equation of circumcircle of triangle $A B P$ .

Important Questions on Ellipse

Learn and Prepare for any exam you want !

A chord of ellipse x2a2+y2b2=1 whose eccentric angles of extremities are α and β, intersects its director circle at point A and B. Tangents at A and B intersect at point P. Find the equation of circumcircle of triangle ABP.

Important Questions on Ellipse

Learn and Prepare for any exam you want !

A chord of ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ whose eccentric angles of extremities are $α$ and $β$ , intersects its director circle at point $A$ and $B$ . Tangents at $A$ and $B$ intersect at point $P$ . Find the equation of circumcircle of triangle $A B P$ .