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Q.
If $x-2 y+k=0$ is a common tangent to $y^2=4 x \quad \& \frac{x^2}{a^2}+\frac{y^2}{3}=1(a >\sqrt{3})$, then the value of a, k and other common tangent are given by -
Conic Sections
Solution:
Given slope of common tangent $m=\frac{1}{2}$.
Equation of general tangent to $y^2=4 x$ is
$y=m x+\frac{1}{m} \ldots \ldots \text { (i) }$
$\Rightarrow y=\frac{1}{2} x+2 \left[\because m=\frac{1}{2} \text { in given equation }\right]$
On comparing with given equation, we get $k =4$
Equation of tangent of $\frac{x^2}{a^2}+\frac{y^2}{3}=1$ is
$y = mx \pm \sqrt{ a ^2 m ^2+3}$.....(ii)
On comparing (i) & (ii)
$\frac{1}{m}=\pm \sqrt{a^2 m^2+3}$.......(iii)
$\Rightarrow a^2=4 \Rightarrow a=\pm 2$.......(iv)
Using (iii) & (iv) we get $m =\pm \frac{1}{2}$.
So equation of other common tangent is $x+2 y+4=0$