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

Answer 1

Answer: (B)

$4y^2=x^2+1$
Point $4y{y}_1=x{x}_1+1$ with $4y_1^2=x_1^2+1$
x axis $\frac{-1}{x},0]$
y axis $\left[0,\frac{1}{4y_1}\right]$
Mid point $h=\frac{-1}{2x},k=\frac{1}{8y_1}$
$x_1=\frac{-1}{2h}{y}_1=\frac{1}{8k}$
$4{\left(\frac{1}{8k}\right)}^2={\left(\frac{-1}{2h}\right)}^2+1$
$\frac{4}{4k^2}=\frac{1}{4b^5}+1$
$\frac{1}{16y^2}=\frac{1}{4b^2}+1$
$\frac{1}{16y^2}=\frac{1+4x^2}{4x^2}$