Ellipse
Circle on Major Axis as Diameter — Triangle Area
nta_pyq_2023_apr
Grade 11
Question:
Let the ellipse $E:\ x^2+9y^2=9$ intersect the positive $x$- and $y$-axes at $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line through $A$ and $B$ meet $C$ at $P$. If the area of $\triangle OAP$ is $\dfrac{m}{n}$ where $m,n$ are coprime, then $m-n$ is equal to
Step-by-Step Solution
Key Concept: $A=(3,0)$, $B=(0,1)$. Circle $C:\ x^2+y^2=9$. Line $AB:\ \frac{x}{3}+y=1$. Find $P$ as second intersection of this line with $C$.
$P$ gives $PQ=\frac{9}{5}$. Area $=\frac{27}{10}$. $m-n=27-10=17$.
Correct Answer: 3