Question

Let a line $L:2x+y=k,k>0$ be a tangent to the hyperbola $x^2-y^2=3.$ If $L$ is also a tangent to the parabola $y^2=\alpha x,$ then $\alpha$ is equal to:

• A. $-24$
• B. $24$
• C. $12$
• D. $-12$

Answer 1

The correct option is A $-24$

Given line is L$:2x+y=k,k>0$
$\Rightarrow L:y=-2x+k$ is tangent to $x^2-y^2=3$
$\therefore x^2-(k-2x{)}^2=3\Rightarrow 3x^2-4kx+k^2+3=0$
Here $\Delta =b^2-4ac=0$
$\therefore k^2=9$
$\because L=0$ is also tangent to $y^2=\alpha x$
$\therefore k=\frac{\alpha /4}{-2}$
$\therefore \alpha =-8k\Rightarrow \alpha =-24$