Question

The length of the latus rectum of the parabola $25[{(x-2)}^2+{(y-4)}^2]={(4x-3y+12)}^2$ is

• A. $\frac{16}{5}$
• B. $\frac{8}{5}$
• C. $\frac{12}{5}$
• D. None of these

Answer 1

The correct option is A $\frac{16}{5}$

The given equation of the parabola can be written as

${(x-2)}^2+{(y-4)}^2={\left(\frac{4x-3y+12}{\sqrt{4^2+{(-3)}^2}}\right)}^2$

$\therefore$ The coordinate of focus are $(2,4)$ and the equation of directrix is $4x-3y+12=0$

The distance of the focus from the directrix $=\frac{|4\left(2\right)-3\left(4\right)+12|}{\sqrt{4^2+{(-3)}^2}}=\frac{8}{5}$

$\therefore$ The length of latus rectum $=2\times \frac{8}{5}=\frac{16}{5}$