Question

If the chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax,$ then the locus of the midpoints of the chords is the curve

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

Answer 1

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

If $(x_1,y_1)$ is the midpoint of the chord to the hyperbola $x^2-y^2=a^2,$
its equation is $T=S_1$
i.e., $x{x}_1-y{y}_1-a^2=x_1^2-y_1^2-a^2$
$\Rightarrow x{x}_1-y{y}_1=x_1^2-y_1^2$
$\Rightarrow y=\frac{x_1}{y_1}x+\frac{y_1^2-x_1^2}{y_1}$

If this is a tangent to $y^2=4ax,$
then $c=\frac{a}{m}$
$\Rightarrow \frac{y_1^2-x_1^2}{y_1}=\frac{a{y}_1}{x_1}$
$\Rightarrow x_1^3=y_1^2(x_1-a)$
$\therefore$ Locus of $(x_1,y_1)$ is $x^3=y^2(x-a)$