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Question #40872

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the surface defined by $z = x|x| + y^2$ . Let $E$ be the set of all points $T(x_0, y_0, z_0)$ on $S$ (where $x_0 \neq 0, y_0 \neq 0$ ) such that the tangent plane to $S$ at $T$ intersects the $x$ -axis at $A(-2, 0, 0)$ and the $y$ -axis at $B(0, 1, 0)$ . Find the number of elements in $E$ .

Options

A.
0
B.
1
C.
2
D.
4

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

Coordinate Axis Projections|Condition of Tangency|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle modulus functions during implicit differentiation of tangency conditions.

A common mistake is to ignore the sign dependency of $x|x|$ and differentiate it as $x^2$ globally, yielding $\frac{\partial z}{\partial x} = 2x$ .

This incorrect assumption leads to the extraneous root $x_0 = -4/5$ , which perfectly satisfies the false gradient condition and leads the student to incorrectly conclude that exactly 1 such point exists.

By enforcing the strict modulus definition, the parity error is exposed, revealing that no such tangency point actually exists.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Coordinate Axis Projections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free