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Q.
$x-2 y+4=0$ is a common tangent to $y^2=4 x \& \frac{x^2}{4}+\frac{y^2}{b^2}=1$. Then the value of $b$ and the other common tangent are given by -
Conic Sections
Solution:
Equation of tangent of ellipse is
$y=m x \pm \sqrt{a^2 m^2+b^2}$.........(i)
Given equation is $x-2 y+4=0$.........(ii)
Since (i) \& (ii) are same, comparing them, we get
$ m =\frac{1}{2} \& \sqrt{ a ^2 m ^2+ b ^2}=2$
$\Rightarrow 4 \cdot \frac{1}{4}+ b ^2=4$
$\Rightarrow b =\pm \sqrt{3}$
Equation of tangent of parabola
$y=m x+\frac{1}{m}$.........(iii)
By (i) & (iii)
$\frac{1}{m^2}=a^2 m^2+b^2$
on solving it we get $m=\pm \frac{1}{2}$
with $m=-\frac{1}{2}$ we get $x+2 y+4=0$
which is other equation of common tangent.