Let C be the centre of the hyperbola frac{x^2 | vedclass

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

Question

(B) Let the point $P$ on the hyperbola be $(a \sec 	heta, b 	an 	heta)$.The equation of the tangent at $P$ is $\frac{x \sec 	heta}{a} - \frac{y

Vedclass · Accepted Answer

$a^2+b^2$

Vedclass · Answer

(B) Let the point $P$ on the hyperbola be $(a \sec 	heta, b 	an 	heta)$.The equation of the tangent at $P$ is $\frac{x \sec 	heta}{a} - \frac{y 	an 	heta}{b} = 1$.The lines are $L_1: bx - ay = 0$ and $L_2: bx + ay = 0$.To find $Q$,solve $\frac{x \sec 	heta}{a} - \frac{y 	an 	heta}{b} = 1$ and $bx = ay \implies y = \frac{bx}{a}$.Substituting $y$: $x(\frac{\sec 	heta}{a} - \frac{	an 	heta}{a}) = 1 \implies x = \frac{a}{\sec 	heta - 	an 	heta} = a(\sec 	heta + 	an 	heta)$.Then $y = b(\sec 	heta + 	an 	heta)$. So $Q = (a(\sec 	heta + 	an 	heta), b(\sec 	heta + 	an 	heta))$.$CQ^2 = a^2(\sec 	heta + 	an 	heta)^2 + b^2(\sec 	heta + 	an 	heta)^2 = (a^2+b^2)(\sec 	heta + 	an 	heta)^2$.Similarly,for $R$,solve with $bx = -ay \implies y = -\frac{bx}{a}$.$x(\frac{\sec 	heta}{a} + \frac{	an 	heta}{a}) = 1 \implies x = \frac{a}{\sec 	heta + 	an 	heta} = a(\sec 	heta - 	an 	heta)$.Then $y = -b(\sec 	heta - 	an 	heta)$. So $R = (a(\sec 	heta - 	an 	heta), -b(\sec 	heta - 	an 	heta))$.$CR^2 = (a^2+b^2)(\sec 	heta - 	an 	heta)^2$.$CQ \cdot CR = \sqrt{(a^2+b^2)(\sec 	heta + 	an 	heta)^2} \cdot \sqrt{(a^2+b^2)(\sec 	heta - 	an 	heta)^2} = (a^2+b^2)|\sec^2 	heta - 	an^2 	heta| = a^2+b^2$.

Let $C$ be the centre of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $P$ be a point on it. If the tangent at $P$ to the hyperbola meets the straight lines $bx-ay=0$ and $bx+ay=0$ respectively in $Q$ and $R$,then $CQ \cdot CR=$

Similar Questions

If $P$ is a point on the hyperbola $16x^2 - 9y^2 = 144$ whose foci are $S_1$ and $S_2$,then $|PS_1 - PS_2| = $

The condition that the straight line $lx + my = n$ may be a normal to the hyperbola $b^2x^2 - a^2y^2 = a^2b^2$ is given by

Find the locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + yk - 4\sqrt{3} = 0$ for different values of $k$.

Let $0 < \theta < \frac{\pi}{2}$. If the eccentricity of the hyperbola $\frac{x^2}{\cos^2 \theta} - \frac{y^2}{\sin^2 \theta} = 1$ is greater than $2$,then the length of its latus rectum lies in the interval

Vedclass Test Series

Exam Paper Generator

Online Exam Module