Question

The equation of common tangent of the parabolas $y^2=8x$ and hyperbola $3x^2-y^2=3$ is

• A. 2x – y + 1 = 0
• B. x – y + 1 = 0
• C. x + y +2 = 0
• D. x + y + 1 = 0

Answer 1

The correct option is A 2x – y + 1 = 0

Equation of tangent to parabola is $y=mx+\frac{2}{m}$
$\Rightarrow mx-y+\frac{2}{m}=0$
It is also tangent to hyperbola $\frac{x^2}{1}-\frac{y^2}{3}=1$
$\Rightarrow 1\left(m^2\right)-3\left(1\right)=\frac{4}{m^2}$
$\Rightarrow m^2-3=\frac{4}{m^2}$
$\Rightarrow m^4-3m^2-4=0$
$\therefore$ equation of common tangent are
$y=\pm 2x\pm 1$
$\Rightarrow y=2x+1$ (or) 2x +y +1 =0