Question

The locus of the mid points of the chords of the hyperbola $x^2-y^2=4,$ which touch the parabola $y^2=8x,$ is

• A. $y^2(x-2)=x^3$
• B. $y^3(x-2)=x^2$
• C. $x^3(x-2)=y^2$
• D. $x^2(x-2)=y^3$

Answer 1

The correct option is A $y^2(x-2)=x^3$

Let mid point of chord of hyperbola $x^2-y^2=4$ be $(x_1,y_1)$
$\therefore$ Equation of chord is:
$x{x}_1-y{y}_1-4=x_1^2-y_1^2-4$
$\therefore y{y}_1=x{x}_1-x_1^2+y_1^2$
$\therefore y=\frac{x_1}{y_1}x+\frac{y_1^2-x_1^2}{y_1}\cdots \left(i\right)$
$\because$ Equation $\left(i\right)$ is tangent to parabola $y^2=8x,$ then
$\frac{y_1^2-x_1^2}{y_1}=\frac{2}{\frac{x_1}{y_1}}$
$\therefore (y_1^2-x_1^2)x_1=2y_1^2$
$\therefore y_1^2(x_1-2)=x_1^3$
$\therefore$ Required locus is: $y^2(x-2)=x^3$