First slide

Parabola

Question

$Let y = 3 x - 8 be the equation of the tangent at the point (7, 13) lying on a parabola whose focus is at (- 1, - 1) . The$
$length of the latus rectum of the parabola is$

Moderate

A

$\frac{20}{\sqrt{13}}$
B

$\frac{20}{13}$
C

$10 \sqrt{13}$
D

$\frac{20}{\sqrt{65}}$

Solution

$The image of focus (- 1, - 1) upon the tangent y = 3 x - 8$
$is the point (5, - 3) . It will lie on the directrix.$
$Slope of directrix = - \frac{7 - 5}{13 + 3} = - \frac{1}{8}$
Its equation is
$y + 3 = - \frac{1}{8} (x - 5)$
$or x + 8 y + 19 = 0$
$∴ Latus rectum = 2 \times (Distance of focus from directrix)$
$= 2 \times \frac{| - 1 - 8 + 19 |}{\sqrt{65}} = \frac{20}{\sqrt{65}}$

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