Question

Equation of a common tangent to the parabola $y^2=4x$ and the hyperbola $xy=2$ is:

• A. $x-2y+4=0$
• B. $x+y+1=0$
• C. $4x+2y+1=0$
• D. $x+2y+4=0$

Answer 1

The correct option is D $x+2y+4=0$

Equation of tangent to $y^2=4x$ is $y=mx+\frac{1}{m}$
This is also tangent to $xy=2$
$x(mx+\frac{1}{m})=2$
$\Rightarrow m{x}^2+\frac{x}{m}-2=0$
For common tangent, $D=0$
${\left(\frac{1}{m}\right)}^2-4\times m(-2)=0$

$\Rightarrow \frac{1}{m^2}+8m=0$

$8m^3+1=0$
$m=-\frac{1}{2}$
$\therefore$ Equation of tangent is $x+2y+4=0$