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Question #40798

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be the set of all values attained by the function $f(x) = \frac{x^2}{2} - 4|x|$ at its points of local extrema, and let $k = n(E)$ be the number of elements in $E$ . For each $m \in E$ , let $L_m$ be the line with slope $m$ which is a valid tangent to the parabola $y^2 = 4x$ . If $L_m$ intersects the latus rectum of the parabola at $Q_m(1, y_m)$ , find the sum of the values of $(m^2 y_m^2 + k)$ over all valid lines $L_m$ .

Options

A.
4227
B.
4226
C.
4230
D.
4228

Check Your Answer

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

Latus Rectum Geometry|Tangent/Normal Slope Interplay|The Modulus Trap|Non-Differentiable Extrema

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of calculus (non-differentiable extrema) and coordinate geometry (tangent conditions).

Students often differentiate $f(x)$ ignoring the modulus, missing the local maximum at $x=0$ , leading to an incorrect count $k=1$ (yielding 4226).

Furthermore, those who do find $m=0$ often blindly substitute it into the algebraic simplification $m^2 y_m^2 = m^2(m + 1/m)^2 = (m^2+1)^2 = 1$ , failing to realize that a parabola $y^2=4ax$ has no valid tangent with slope $m=0$ .

This 'Lethal Collision' of blindly applying formulas without geometric validation leads to the trap answer 4230, effectively separating top candidates.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Latus Rectum Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free