Equation of a common tangent to the parabola $y^{2}=4 x$ and the hyperbola $x y=2$ is :
Equation of a common tangent to the parabola $y^{2}=4 x$ and the hyperbola $x y=2$ is :
$x+y+1=0$
$x-2 y+4=0$
$x+2 y+4=0$
$4 x+2 y+1=0$
Solution
Equation of a tangent to parabola $y^{2}=4 x$ is:
$y=m x+\frac{1}{m}$
This line is a tangent to $x y=2$ $\therefore \quad x\left(m x+\frac{1}{m}\right)=2 \Rightarrow m x^{2}+\frac{1}{m} x-2=0$
$\because$ Tangent is common for parabola and hyperbola. $\therefore \quad D=\left(\frac{1}{m}\right)^{2}-4 \cdot m \cdot(-2)=0$
$\frac{1}{m^{2}}+8 m=0$
$1+8 m^{3}=0$
$m^{3}=-\frac{1}{8} \Rightarrow m=-\frac{1}{2}$
$\therefore$ Equation of common tangent: $y=-\frac{1}{2} x-2$
$\Rightarrow 2 y=-x-4 \Rightarrow x+2 y+4=0$