JEE Main · 2024 · 05 Apr 2024 · Shift 1 · Ellipse

Let the line

2 x + 3 y - k = 0, k > 0

, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is

x^{2} + y^{2} - 3 x - 2 y = 0

and the length of the latus rectum of the ellipse

x^{2} + 9 y^{2} = k^{2}

\frac{m}{n}

, where

m

and

n

are coprime, then

2 m + n

is equal to

Answer: (D)

Solution

The circle

x^{2} + y^{2} - 3 x - 2 y = 0

has centre

(\frac{3}{2}, 1)

, which is the midpoint of

A B

, so it lies on the diameter line

2 x + 3 y - k = 0

2 \cdot \frac{3}{2} + 3 \cdot 1 - k = 0 \Rightarrow k = 6

.
Ellipse:

x^{2} + 9 y^{2} = 36 \Rightarrow \frac{x ^{2}}{36} + \frac{y ^{2}}{4} = 1

a = 6, b = 2

. Latus rectum

= \frac{2 b ^{2}}{a} = \frac{8}{6} = \frac{4}{3} = \frac{m}{n}

m = 4, n = 3

.
Thus

2 m + n = 8 + 3 = 11.

JEE Main · 2024 · 05 Apr 2024 · Shift 1 · Single correct option