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Question #40803

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ and $S(4,0)$ be its focus. A variable tangent $L$ touches $E$ at the point $P(5\cos\theta, 3\sin\theta)$ for $\theta \in [0, 2\pi)$ . Let $Q(x,y)$ be the orthogonal projection of $S$ onto $L$ . Find the number of values of $\theta$ such that the coordinates of $Q$ satisfy the constraint $|x| + |y| = 7$ .

Options

A.
4
B.
6
C.
8
D.
10

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Ellipse Auxiliary Circle|Orthogonal projection of a line|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

The core trap is the 'Sign Flipping Rule' applied to the modulus constraint $|20+25\cos\theta|$ .

Students often blindly assume $20+25\cos\theta > 0$ because they associate it with the positive radius or standard first-quadrant geometry, missing the domain where $\cos\theta < -4/5$ .

This forces a sign flip in the projection vector's $x$ -component, creating a hidden secondary equation ( $15|\sin\theta| = 55+53\cos\theta$ ) that yields 4 additional valid solutions.

Failing to partition the domain based on the modulus argument leads to a 50% undercounting of the roots.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Ellipse Auxiliary Circle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free