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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

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Detailed Solution

$Equation of AB is x + 3 y = 3 and circle is x^{2} + y^{2} = 9 Solve above y = 0, y = \frac{9}{5} Area = \frac{1}{2} \times 3 \times \frac{9}{5} = \frac{27}{10} = \frac{m}{n}$

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