Ellipse 2 Question 7
7. Equation of a common tangent to the parabola $y^{2}=4 x$ and the hyperbola $x y=2$ is
(2019 Main, 11 Jan I)
(a) $x+2 y+4=0$
(b) $x-2 y+4=0$
(c) $4 x+2 y+1=0$
(d) $x+y+1=0$
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Solution:
- We know that, $y=m x+\frac{a}{m}$ is the equation of tangent to the parabola $y^{2}=4 a x$.
$\therefore y=m x+\frac{1}{m}$ is a tangent to the parabola
$y^{2}=4 x$.
$[\because a=1]$
Let, this tangent is also a tangent to the hyperbola $x y=2$
Now, on substituting $y=m x+\frac{1}{m}$ in $x y=2$, we get
$$ \begin{aligned} x m x+\frac{1}{m} & =2 . \\ \Rightarrow \quad m^{2} x^{2}+x-2 m & =0 \end{aligned} $$
Note that tangent touch the curve exactly at one point, therefore both roots of above equations are equal.
$$ \begin{aligned} & \Rightarrow \quad D=0 \Rightarrow 1-4\left(m^{2}\right)(-2 m) \Rightarrow m^{3}=-\frac{1}{2} \\ & \Rightarrow \quad m=-\frac{1}{2} \end{aligned} $$
$\therefore$ Required equation of tangent is
$$ \begin{aligned} & & y & =-\frac{x}{2}-2 \\ \Rightarrow & & 2 y & =-x-4 \\ \Rightarrow & & x+2 y+4 & =0 \end{aligned} $$
