If the tangents drawn to the hyperbola 4y^2 = | vedclass

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

Question

(D) The equation of the hyperbola is $4y^2 - x^2 = 1$.Let the point of tangency be $P(x_1, y_1)$. The equation of the tangent at $P$ is $4yy_1 - xx_1

Vedclass · Accepted Answer

$x^2 - 4y^2 - 16x^2y^2 = 0$

Vedclass · Answer

(D) The equation of the hyperbola is $4y^2 - x^2 = 1$.Let the point of tangency be $P(x_1, y_1)$. The equation of the tangent at $P$ is $4yy_1 - xx_1 = 1$.This tangent intersects the $x$-axis at $A$ where $y=0$,so $-xx_1 = 1 \Rightarrow x = -1/x_1$. Thus,$A = (-1/x_1, 0)$.This tangent intersects the $y$-axis at $B$ where $x=0$,so $4yy_1 = 1 \Rightarrow y = 1/(4y_1)$. Thus,$B = (0, 1/(4y_1))$.Let the midpoint of $AB$ be $(h, k)$.Then $h = -1/(2x_1) \Rightarrow x_1 = -1/(2h)$ and $k = 1/(8y_1) \Rightarrow y_1 = 1/(8k)$.Since $(x_1, y_1)$ lies on the hyperbola $4y_1^2 - x_1^2 = 1$,we substitute the values:$4(1/(8k))^2 - (-1/(2h))^2 = 1$$4/(64k^2) - 1/(4h^2) = 1$$1/(16k^2) - 1/(4h^2) = 1$Multiplying by $16h^2k^2$:$h^2 - 4k^2 = 16h^2k^2$Replacing $(h, k)$ with $(x, y)$,the locus is $x^2 - 4y^2 - 16x^2y^2 = 0$.

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the coordinate axes at the distinct points $A$ and $B$,then the locus of the midpoint of $AB$ is

Similar Questions

The foci of the hyperbola $2x^2 - 3y^2 = 5$ are

If the angle between the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{3} = 4$ is $\frac{\pi}{3}$,then its conjugate hyperbola is:

Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 3)$,foci $(0, \pm 5)$.

$A$ normal to the hyperbola $4x^2 - 9y^2 = 36$ meets the coordinate axes $x$ and $y$ at $A$ and $B$,respectively. If the parallelogram $OABP$ ($O$ being the origin) is formed,then the locus of $P$ is

If a circle cuts a rectangular hyperbola $xy = c^2$ at four points $A, B, C,$ and $D$,and the parameters of these four points are $t_1, t_2, t_3,$ and $t_4$ respectively,then which of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module