Question

The length of the minor axis of the ellipse ${\left(5x-10\right)}^2+{\left(5y+15\right)}^2=\frac{{\left(3x-4y+7\right)}^2}{4}$ is

• A. $\frac{10}{3}$ units
• B. $\frac{10}{\sqrt{3}}$ units
• C. $\frac{20}{3}$ units
• D. $\frac{5}{\sqrt{3}}$ units

Answer 1

Answer: (B)

${\left(x-2\right)}^2+{\left(y+3\right)}^2={\left(\frac{1}{2}\frac{3x-4y+7}{5}\right)}^2$ is an ellipse, whose focus is $\left(2,-3\right),$ directix $3x-4y+7=0$ and eccentricity is $\frac{1}{2}$
Length of $\perp$ 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, $2b=\frac{10}{\sqrt{3}}$