Let the ellipse intersect the positive - and -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 with vertices

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Solution

Q.
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 with vertices A, P and the origin O is $\frac{m}{n}$ , where m and n are coprime, then $m - n$ is equal to [2023]

1 17

2 15

3 18

4 16

Ans.
(1)

We have, $E : \frac{x^{2}}{9} + \frac{y^{2}}{1} = 1$

$A \equiv (3, 0)$

$B \equiv (0, 1)$

Equation of line passing through A(3, 0) and B(0, 1) is $\frac{y - 0}{x - 3} = \frac{1 - 0}{0 - 3} \Rightarrow x + 3 y = 3$ ...(i)

The equation of the circle with radius 3 is $x^{2} + y^{2} = 9$ ...(ii)

From (i) and (ii), we get
${(3 - 3 y)}^{2} + y^{2} = 9 \Rightarrow 10 y^{2} - 18 y = 0$ $\Rightarrow y = 0, \frac{9}{5}$

$∴$ $Area of triangle O P A = \frac{1}{2} \times 3 \times \frac{9}{5} = \frac{27}{10} = \frac{m}{n}$

$∴$ $m - n = 17$

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