Question

The length of the latus-rectum of the parabola $169\left\{\right(x-1{)}^2+(y-3{)}^2\}=(5x-12y+17{)}^2$ is

• A. $\frac{12}{13}$
• B. $\frac{14}{13}$
• C. $\frac{28}{13}$
• D. $\frac{31}{13}$

Answer 1

The correct option is C $\frac{28}{13}$

The given equation can be written as $(x-1{)}^2+(y-3{)}^2$=${\left(\frac{5x-12y+17}{13}\right)}^2$
This is of the form PS = PM, where P is the point (x,y) on the parabola, PM is the perpendicular distance of P from the line 5x-12y+17 = 0(directrix) and PS is the of P from focus (1,3)
So we get focus = (1,3) and directrix equal to 5x-12y+17 = 0.
Length of latus rectum =4a
Perpendicular distance from (1,3) to the line 5x-12y+17=0 is
$2a=\frac{|5\times 1-12\times 3+17|}{\sqrt{169}}=\frac{14}{13}$
Answer = 4a = $\frac{28}{13}$