Question

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

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

Answer 1

The correct option is D

$y=x+2$

Equation of tangent to $y^2=8x$ is $y=mx+\frac{2}{m}\dots \left(1\right)$

But it touches $xy=-1\Rightarrow x(mx+\frac{2}{m})=-1$ must have equal roots.

$\Rightarrow m{x}^2+\frac{2}{m}x+1=0$
$\Rightarrow \Delta =\frac{4}{m^2}-4m=0\Rightarrow 1-m^3=0\Rightarrow m=1\therefore \left(1\right)\Rightarrow y=x+2$