Let C be the centre of an ellipse and PQ be a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let $P = (a \cos 	heta, b \sin 	heta)$ and $Q = (a \cos \phi, b \sin

Vedclass · Accepted Answer

an ellipse

Vedclass · Answer

(C) Let the equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let $P = (a \cos 	heta, b \sin 	heta)$ and $Q = (a \cos \phi, b \sin \phi)$. Since $\angle PCQ = 90^{\circ}$,the product of the slopes of $CP$ and $CQ$ is $-1$. Thus,$\frac{b \sin 	heta}{a \cos 	heta} 	imes \frac{b \sin \phi}{a \cos \phi} = -1$,which implies $\frac{b^2}{a^2} 	an 	heta 	an \phi = -1$,or $	an 	heta 	an \phi = -\frac{a^2}{b^2}$. The intersection point $R(h, k)$ of the tangents at $P$ and $Q$ is given by $h = a \frac{\cos(\frac{	heta+\phi}{2})}{\cos(\frac{	heta-\phi}{2})}$ and $k = b \frac{\sin(\frac{	heta+\phi}{2})}{\cos(\frac{	heta-\phi}{2})}$. Squaring and rearranging,we find that $\frac{h^2}{a^4} + \frac{k^2}{b^4} = \frac{1}{a^2 \cos^2(\frac{	heta-\phi}{2})} + \frac{1}{b^2 \cos^2(\frac{	heta-\phi}{2})}$. Using the condition $	an 	heta 	an \phi = -\frac{a^2}{b^2}$,it can be shown that the locus of $R$ is $\frac{x^2}{a^4} + \frac{y^2}{b^4} = \frac{1}{a^2} + \frac{1}{b^2}$,which represents another ellipse.

Let $C$ be the centre of an ellipse and $PQ$ be a chord of it with $\angle PCQ = 90^{\circ}$. If $R$ is the point of intersection of the tangents to the ellipse at $P$ and $Q$,then $R$ lies on

Similar Questions

$A$ point on the curve $x=3 \cos \theta, y=2 \sin \theta$ at which the tangent is perpendicular to the $X$-axis is

The locus of the point of intersection of the perpendicular tangents to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is

The eccentricity of an ellipse whose latus rectum is equal to the distance between its two foci is:

Find the equation of the chord of the ellipse $2x^2 + 5y^2 = 20$ which is bisected at the point $(2, 1)$.

Consider the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

Vedclass Test Series

Exam Paper Generator

Online Exam Module