Ellipse
Position of a Point
IIT-JEE 1994
Grade 11
Question:
Let $E$ be the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and $C$ be the circle $x^2 + y^2 = 9$. Let $P = (1, 2)$ and $Q = (2, 1)$. Then:
(1) $Q$ lies inside $C$ but outside $E$
(2) $Q$ lies outside both $C$ and $E$
(3) $P$ lies inside both $C$ and $E$
(4) $P$ lies inside $C$ but outside $E$
Step-by-Step Solution
Key Concept: For $P$: $1/9 + 4/4 = 10/9 > 1$ (outside $E$); $1 + 4 = 5 < 9$ (inside $C$) $\Rightarrow$ (4). For $Q$: $4/9 + 1/4 = 25/36 < 1$ (inside $E$); $4 + 1 < 9$ (inside $C$).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (4)