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Question #41358

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = x \sin\left(\frac{\pi x}{2}\right)$ for $x \in [0, 1]$ . By Lagrange's Mean Value Theorem, there exists $c \in (0, 1)$ such that the tangent $T$ to the curve $y = f(x)$ at $x = c$ is parallel to the chord joining the endpoints of the curve.

A curve $C$ in the complex plane is defined by the locus of $z = x + iy$ satisfying $|z - 5| - |z + 5| = 8$ .

If a normal to the curve $C$ is parallel to $T$ , then the equation of this normal is:

Options

A.
$y = x + \frac{25}{\sqrt{7}}$
B.
$y = x - \frac{25}{\sqrt{7}}$
C.
$y = x \pm \frac{25}{\sqrt{7}}$
D.
$y = x + \sqrt{7}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

LMVT Tangent Parallelism|Normal to Conic|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of differential calculus (LMVT) and coordinate geometry, featuring a critical trap in the complex plane locus definition.

Students often mechanically convert the modulus equation $|z - 5| - |z + 5| = 8$ into the full hyperbola and find normals for both branches, leading to the incorrect selection of the dual-equation option.

The 'Geometrical Interpretation of Modulus' strictly restricts the domain to the left branch ( $x < 0$ ) because the distance to the right focus is strictly greater than the distance to the left focus.

Failing to apply this domain restriction results in extraneous solutions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	LMVT Tangent Parallelism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free