Question

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