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Question #40701

Difficulty: 8/10 | 🟢 Free

The Question

Let $H$ be the harmonic mean of the roots of the polynomial equation $x^3 - 11x^2 + 36x - 36 = 0$ . Consider the curve $C: y = 2x^2 - (3H+2)x + H^2 + 5H$ and the line $L: y = (H-2)x + c$ . If the $x$ -coordinates of the intersection points of $C$ and $L$ are roots of an equation with even multiplicity, find the value of $x_0 + y_0 + c$ , where $(x_0, y_0)$ is the point of contact.

Options

A.
12
B.
15
C.
18
D.
24

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

Harmonic Mean Constraints|Parabola Shifted Vertex|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the theory of equations (Harmonic Mean via Vieta's formulas) and coordinate geometry (tangency conditions).

The primary trap lies in the 'even multiplicity' phrasing.

Students often default to solving for two distinct intersection points when equating a line and a parabola.

The phrase dictates that the intersection quadratic must be a perfect square ( $\Delta = 0$ ), geometrically representing tangency at the HM point.

Furthermore, calculating the HM requires recognizing $H = 3\alpha\beta\gamma / \sum\alpha\beta$ rather than explicitly finding the roots, which saves crucial time.

Missing the multiplicity condition leaves the system underdetermined, preventing the discovery of $c$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Harmonic Mean Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free