JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Ellipse

Let the ellipse

E : x^{2} + 9 y^{2} = 9

intersect the positive x - and y -axes at the points A and B respectively Let the major axis of E be a diameter of the circle C . Let the line passing through A and B meet the circle C at the point P . If the area of the triangle which vertices

A, P

and the origin O is

\frac{m}{n}

, where m and n are coprime, then

m - n

is equal to

(A)18
(B)16
(C)17✓ Correct
(D)15

Answer: (C)

Solution

The ellipse

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

meets the positive axes at

A (3, 0)

and

B (0, 1)

. Its major axis has length

6

and is a diameter of circle

C

, so

C

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

. The line

A B

x + 3 y = 3

. Substituting

x = 3 - 3 y

into

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

gives

10 y^{2} - 18 y = 0

, so

y = 0

(point

A

) or

y = \frac{9}{5}

, giving

P (- \frac{12}{5}, \frac{9}{5})

. Taking

O A

along the

x

-axis as base, its length is

3

and the corresponding height is the ordinate of

P

, namely

\frac{9}{5}

. The area of

△ O A P

\frac{1}{2} \cdot 3 \cdot \frac{9}{5} = \frac{27}{10}

, so

m = 27

n = 10

and

m - n = 17

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Single correct option

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JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Ellipse

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