Question

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$ , then the locus of the mid point of $AB$ is :

• A. $x^2 - 4y^2 + 16x^2y^2 = 0$
• B. $x^2 - 4y^2 - 16x^2y^2 = 0$
• C. $4x^2 - y^2 + 16x^2y^2 = 0 $
• D. $4x^2 - y^2 - 16x^2y^2 = 0$

Answer 1

Answer: (B)

$4 y^{2}=x^{2}+1 $
Point $ 4 y y_{1}=x x_{1}+1$ with $4 y_{1}^{2}=x_{1}^{2}+1$
x axis $\frac{-1}{x}, 0] $
y axis $\left[0, \frac{1}{4 y_{1}}\right] $
Mid point $h=\frac{-1}{2 x}, k=\frac{1}{8 y_{1}}$
$x_{1}=\frac{-1}{2 h} y_{1}=\frac{1}{8 k} $
$4\left(\frac{1}{8 k}\right)^{2}=\left(\frac{-1}{2 h}\right)^{2}+1 $
$\frac{4}{4 k^{2}}=\frac{1}{4 b^{5}}+1 $
$\frac{1}{16 y^{2}}=\frac{1}{4 b^{2}}+1 $
$\frac{1}{16 y^{2}}=\frac{1+4 x^{2}}{4 x^{2}}$