Parabola 2 Question 12
12. The equation of the common tangent to the curves $y^{2}=8 x$ and $x y=-1$ is
(2002, 1M)
(a) $3 y=9 x+2$
(b) $y=2 x+1$
(c) $2 y=x+8$
(d) $y=x+2$
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Solution:
- Tangent to the curve $y^{2}=8 x$ is $y=m x+\frac{2}{m}$. So, it must satisfy $x y=-1$
$$ \Rightarrow \quad x m x+\frac{2}{m}=-1 \Rightarrow m x^{2}+\frac{2}{m} x+1=0 $$
Since, it has equal roots.
$$ \begin{array}{lrl} \therefore & D=0 \\ \Rightarrow & \frac{4}{m^{2}}-4 m & =0 \\ \Rightarrow & m^{3} & =1 \\ \Rightarrow & m & =1 \end{array} $$
Hence, equation of common tangent is $y=x+2$.
