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Let the ellipse E : $\begin{aligned} & x^2+9y^2=9 \\ \end{aligned}$ 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 $\begin{aligned} & \frac{m}{n} \end{aligned}$ where m and n are coprime, then m – n is equal to :
Option: 1
16

Option: 2
15

Option: 3
18

Option: 4
17

Answers (1)

best_answer

Equation of line AB or AP is

$\frac{x}{3}+\frac{y}{1}=1\\$

$x+3y=3$

$x=(3-3y)$

Intersection point of line AP & circle is $P(x_{0},y_0)$

$$$ \begin{aligned} & x^2+y^2=9 \Rightarrow(3-3 y)^2+y^2=9 \\ & \Rightarrow 3^2\left(1+y^2-2 y\right)+y^2=9 \\ & \Rightarrow 5 y^2-9 y=0 \Rightarrow y(5 y-9)=0 \\ & \Rightarrow y=9 / 5 \end{aligned}$

Hence, $x=3(1-y)=3\left(1-\frac{9}{5}\right)=3\left(\frac{-4}{5}\right)$

$x=\frac{-12}{5}$

$P\left(x_0, y_0\right)=\left(\frac{-12}{5}, \frac{9}{5}\right)$

Area of $\Delta AOP$ is $\frac{1}{2}\times OA\times Height$

Height = 9 / 5 , OA =3

$=\frac{1}{2}\times 3\times \frac{9}{5}=\frac{27}{10}=\frac{m}{n}$

Compare both side $m=27,$ $n=10,$ $\Rightarrow m-n=17$

Therefore, correct answer is option-D

Posted by

Ritika Harsh

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