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Q.
The length of the minor axis of the ellipse $\left(5 x - 10\right)^{2}+\left(5 y + 15\right)^{2}=\frac{\left(3 x - 4 y + 7\right)^{2}}{4}$ is
NTA AbhyasNTA Abhyas 2022
Solution:
$\left(x - 2\right)^{2}+\left(y + 3\right)^{2}=\left(\frac{1}{2} \, \frac{3 x - 4 y + 7}{5}\right)^{2}$ is an ellipse, whose focus is $\left(2 , - 3\right),$ directix $3x-4y+7=0$ and eccentricity is $\frac{1}{2}$
Length of $\bot$ from focus to directrix is $\frac{3 \times 2 - 4 \times \left(- 3\right) + 7}{5}=5$
$\frac{a}{e}-ae=5\Rightarrow 2a-\frac{a}{2}=5\Rightarrow a=\frac{10}{3}$ and $b=\frac{5}{\sqrt{3}}$
So, length of major axis is $\frac{20}{3}$
and length of minor axis, $2b=\frac{10}{\sqrt{3}}$