Question:
Equation of a common tangent to the parabola $y^{2}=4 x$ and the hyperbole $x y=2$ is :
Correct Option: 1
Solution:
Let the equation of tangent to parabola
$y^{2}=4 x$ be $y=m x+\frac{1}{m}$
It is also a tangent to hyperbola $x y=2$
$\Rightarrow x\left(m x+\frac{1}{m}\right)=2$
$\Rightarrow x^{2} m+\frac{x}{m}-2=0$
$\mathrm{D}=0 \Rightarrow \mathrm{m}=-\frac{1}{2}$
So tangent is $2 y+x+4=0$