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

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Solution

The correct option is C
The given equation can be written as $(x - 1)^{2} + (y - 3)^{2}$ = ${(\frac{5 x - 12 y + 17}{13})}^{2}$
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
$2 a = \frac{| 5 \times 1 - 12 \times 3 + 17 |}{\sqrt{169}} = \frac{14}{13}$
Answer = 4a = $\frac{28}{13}$

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