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10-2. Parabola, Ellipse, Hyperbola
hard
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
$17$
D
$15$
(JEE MAIN-2023)
Solution
For line $AB x+3 y =3$ and circle is $x ^2+ y ^2=9$
$(3-3 y)^2+y^2=9$
$\Rightarrow 10 y^2-18 y=0$
$\Rightarrow y=0, \frac{9}{5}$
$\therefore \text { Area }=\frac{1}{2} \times 3 \times \frac{9}{5}=\frac{27}{10}$
$m – n =17$
Standard 11
Mathematics
