Question

The equation of the common tangent to the curves $y^2=8x$ and $xy=-1$ is

• A. 3y = 9x + 2
• B. y = 2x + 1
• C. 2y =x +8
• D. y =x +2

Answer 1

Answer: (D)

Tangent to the curve $y^2=8x$ is $y=mx+\frac{2}{m}$.
So, it must satisfy $xy=-1$
$\Rightarrow x\left(mx+\frac{2}{m}\right)=-1$
$\Rightarrow m{x}^2+\frac{2}{m}x+1=0$
Since, it has equal roots.
$\therefore D=0$
$\Rightarrow \frac{4}{m^2}-4m=0$
$\Rightarrow m^3=1$
$\Rightarrow m=1$
So, equation of common tangent is $y=x+2$.