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Question #40808

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all values of $a^2$ for which the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{9} = 1$ is tangent to the curve $C: y = 2|x - 3| + 1$ . The sum of the elements of $S$ is:

Options

A.
4
B.
10
C.
14
D.
$\frac{81}{8}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Condition of Tangency|Tangent/Normal Slope Interplay|Non-Differentiable Extrema

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to verify the domain of tangency when dealing with piecewise functions.

The 'Lethal Collision' occurs in two ways: First, students might blindly apply the standard analytic condition of tangency ( $c^2 = a^2m^2 + b^2$ ) to both branches, yielding $a^2 = 4$ and $a^2 = 10$ , and sum them to 14 without checking if the point of contact lies on the valid domain of the respective branch.

Second, students might assume tangency occurs exactly at the non-differentiable vertex $(3,1)$ , leading to the erroneous $a^2 = 81/8$ .

The problem forces students to recognize that algebraic tangency to a line does not guarantee geometric tangency to a restricted ray.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Condition of Tangency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free