Thank you for reporting, we will resolve it shortly
Q.
The length of major axis of the ellipse $(5 x-10)^2+(5 y+15)^2 =\frac{(3 x-4 y+7)^2}{4}$ is:
Conic Sections
Solution:
$(x-2)^2+(y+3)^2=\left(\frac{1}{2}\right)^2\left(\frac{3 x-4 y+7}{5}\right)^2$
is an ellipse, whose focus is $(2,-3)$, directrix $3 x-4 y+7=0$ and eccentricity $\frac{1}{2}$.
Length of the perpendicular from the focus to the directrix is $\frac{3 \times 2-4 \times(-3)+7}{5}=5$ so that $\frac{a}{e}-a e=5$
$\Rightarrow 2 a-\frac{a}{2}=5 \Rightarrow a=\frac{10}{3}$
So length of the major axis is $\frac{20}{3}$