Question

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

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

Answer 1

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

$y^2=8x$
Equation of tangent to this parabola $y^2=4ax$ is:
$y=mx+\frac{a}{m}$
$\Rightarrow y=mx+\frac{2}{m}\cdots \left(i\right)$
Solving the equation of tangent with the curve $xy=-1$
$\Rightarrow m{x}^2+\frac{2x}{m}=-1$
$\Rightarrow m^2{x}^2+2x+m=0$
Quadratic will have equal roots $\Rightarrow D=0$
$\Rightarrow 4-4m^3=0$
$\Rightarrow m=1$
Putting value of m in $\left(i\right)$
Equation of common tangent is
$\Rightarrow y=x+2$