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

Find the coordinates of the foci and the vertices,the eccentricity,and the length of the latus rectum of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$.

The eccentricity of the conic section $x^2 - 4y^2 = 1$ is:

Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be $(\pm 5, 0)$ and $\sqrt{50}$,respectively. Then,the square of the eccentricity of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2b^2}=1$ equals

$A$ hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$. Then the tangent to this hyperbola at $P$ also passes through the point:

If the line $y=mx+c$ is a common tangent to the hyperbola $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36,$ then which one of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module