Let the ellipse E: x ^ 2 + 9 y ^ 2 = 9 intersect the positive x -...

Given,

Ellipse $E: {x}^{2} + 9{y}^{2} = 9 . . . . . . . . \left(i\right)$

Now point $A\left(3, 0\right) \text{ and } B\left(0, 1\right)$ which is the intersection of given ellipse with positive axis,

So, equation of line passing through $A \text{ and } B$ is given by,

$L: \dfrac{x}{3} + \dfrac{y}{1} = 1 ⇒ x = 3 - 3y . . . . . . . \left(ii\right)$

Now equation of Circle with diametric point \left(- 3, 0\right) \text{ and } \left(3, 0\right)$is given by,</p><p>$C: {x}^{2} + {y}^{2} = 9 . . . . . . . . \left(iii\right)$</p><p>Let$Q$be foot of perpendicular from$P$upon major axis,</p><p>So, from \left(ii\right) \text{ and } \left(iii\right)$ we get,

${\left(3 - 3y\right)}^{2} + {y}^{2} = 9$

$⇒ y = \dfrac{9}{5}, 0$

Hence, $PQ = \dfrac{9}{5}$

Now Area of triangle will be,

$= \dfrac{1}{2} \times OA \times PQ = \dfrac{1}{2} \times 3 \times \dfrac{9}{5} = \dfrac{27}{10}$

Hence, $m - n = 17$