Question

Equation of a common tangent to the curves $y^2=8x$ and $xy=-1$ is:

• A. $y=4x-4$
• B. $y=2x+1$
• C. $2y=8+x$
• D. $y=x+2$

Answer 1

The correct option is A $y=4x-4$

Given equations of the curves are $y^2=8x$ and $xy=-1$

Finding the point of intersection of the two curves gives,

$y^3=-8⟹y=-2⟹x=\frac{1}{2}$

$\therefore$ The point of intersection is $(\frac{1}{2},-2)$,

The slope of the tangent at that point is $\frac{dy}{dx}=\frac{1}{x^2}=4$,

$\therefore$ The equation of the common tangent is $y+2=4(x-\frac{1}{2})$

$⟹y+2=4x-2$

$⟹y-4x+4=0$