If the line x - 2y = 12 is a tangent to the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 = 1 at the point (3, - (9)/(2)) , then the length of the latus rectum of the ellipse is

Correct answer: C. 9 units ∵ ( 3, - (9)/(2)) lies on x^ 2 a^ 2 + y^ 2 b^ 2 = 1 ⇒ 9 a^ 2 + 81 4b^ 2 = 1 ……(1) Equation of the tangent at ( 3, - (9)/(2)) is 3x a^ 2 + - (9)/(2)y b^ 2 = 1 & given equation of the tangent is: x - 2y = 12 ⇒ (x)/(12) + (- y)/(6) = 1 On comparing these equations: a^ 2 3 = 12 ⇒ a^ 2 = 36 ⇒ a = 6 2b^ 2 9 = 6 ⇒ b^ 2 = 27 ⇒ b = 3 3 ∴ LengthofL . R . = 2b^ 2 a = (2 × 27)/(6) = 9

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Ellipse

NTA Abhyas 2020 · Mathematics · Ellipse

If the line x - 2y = 12 is a tangent to the ellipse x^ 2 a^ 2 + y^...

If the line

x - 2y = 12

is a tangent to the ellipse

\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1

at the point \left(3, - \dfrac{9}{2}\right)$, then the length of the latus rectum of the ellipse is

Options

A $5$ units
B $12\sqrt{2}$ units
C $9$ units
D $8\sqrt{3}$ units

Correct answer

C. $9$ units

Step-by-step solution

∵ \left(

3, -

\dfrac{9}{2}\right) lies on \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} =

1 ⇒

\dfrac{9}{a^{2}} + \dfrac{81}{4b^{2}} =

&hellip;&hellip;(1)<br />Equation of the tangent at \left(

3, -

\dfrac{9}{2}\right) is \dfrac{3x}{a^{2}} + \dfrac{- \dfrac{9}{2}y}{b^{2}} =

<br />& given equation of the tangent is:

x - 2y = 12 ⇒

\dfrac{x}{12} + \dfrac{- y}{6} =

<br />On comparing these equations: \dfrac{a^{2}}{3} =

12 ⇒ a^{2} = 36 ⇒ a = 6

<br />\dfrac{2b^{2}}{9} =

6 ⇒ b^{2} = 27 ⇒ b = 3

\sqrt{3}<br />

∴ LengthofL . R . =

\dfrac{2b^{2}}{a} = \dfrac{2 \times 27}{6} =

Practice Ellipse on Quantrex Academy →

More from Ellipse