Q42 / 53 · NTA Abhyas (JEE Main) 2020 · Mathematics · Ellipse

The length of the latus rectum of the curve represented by $x = 3 (\cos t + \sin t)$ and $y = 4 (\cos t - \sin t),$ is

A
$\frac{9}{2}$ units
B
$9 \sqrt{2}$ units
C
$\frac{32 \sqrt{2}}{3}$ units
Correct answer
D
$\frac{9}{\sqrt{2}}$ units

Solution

\frac{x}{3} = \cos t + \sin t

and

\frac{y}{4} = \cos t - \sin t

{(\frac{x}{3})}^{2} + {(\frac{y}{4})}^{2} = 2

\frac{x^{2}}{{(3 \sqrt{2})}^{2}} + \frac{y}{{(4 \sqrt{2})}^{2}} = 1

Hence, the length of the latus rectum

= \frac{2 {(3 \sqrt{2})}^{2}}{4 \sqrt{2}} = \frac{9}{\sqrt{2}}

Hence, the correct answer is option: 'D'.

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