Ellipse
Auxiliary Circle
IIT-JEE 2009 Paper 1
Grade 11
Question:
The line through extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^2 + 9y^2 = 9$ meets its auxiliary circle at the point $M$. The area of the triangle with vertices at $A$, $M$, and the origin $O$ is:
(1) $\frac{31}{10}$
(2) $\frac{29}{10}$
(3) $\frac{21}{10}$
(4) $\frac{27}{10}$
Step-by-Step Solution
Key Concept: $A = (3, 0)$, $B = (0, 1)$; auxiliary circle $x^2+y^2 = 9$. Line $AB$: $x+3y = 3$. Second intersection $M$ with $x^2+y^2 = 9$: solve to get $M = (-12/5, 9/5)$. Area of $\triangle OAM = \frac{1}{2}|3\cdot 9/5 - 0| = 27/10$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (4)