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)

Any tangent to the parabola $y^2=8x$ is
$y=mx+\frac{2}{m}$.
Clearly (a), (b) and (d) satisfy the equation (by $m=$ $3,2,1)$.
For $xy=-1$
Equation of tangent with slope $m$, to $xy=-1$ are
$y-\sqrt{m}=m\left(x+\frac{1}{\sqrt{m}}\right)$
$\left(\because \frac{dy}{dx}=\frac{1}{x^2}=m\therefore x=\pm \frac{1}{\sqrt{m}},y=\mp \sqrt{m}\right)$
Clearly (d) satisfies these equations.