Question

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

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

Answer 1

The correct option is D $y=x+2$

Tangent equation to $y^2=8x$ is $y=mx+\frac{2}{m}$
Solving with $xy=-1,$we get
$x(mx+\frac{2}{m})=-1$
$\Rightarrow m{x}^2+\frac{2}{m}x+1=0$
Discriminant must be $0$ because line is also tangent to $xy=-1$
$\therefore {\left(\frac{2}{m}\right)}^2-4m=0$
$\Rightarrow {\left(\frac{2}{m}\right)}^2=4m$
$\Rightarrow m^3=1\Rightarrow m=1$
$\therefore$ tangent equation is $y=x+2$