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Question #73601

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u} = x \hat{i}$ and $\vec{v} = y \hat{j}$ (where $x, y > 0$ ) be two vectors. The line passing through the endpoints of $\vec{u}$ and $\vec{v}$ is tangent to the auxiliary circle of the ellipse $\frac{X^2}{9} + \frac{Y^2}{4} = 1$ . Let $S$ be the area of the parallelogram defined by $\vec{u}$ and $\vec{v}$ . Find the sum of all possible integer values of $\frac{x^2}{y^2}$ that satisfy the inequality $\frac{S - 36}{S - 18} \le 0$

Options

A.
90
B.
91
C.
78
D.
105

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

Geometric Interpretation of Cross Product|Ellipse Auxiliary Circle|Wavy Curve (Method of Intervals)|Denominator Restriction

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the geometric interpretation of the cross product and tangency conditions to an ellipse's auxiliary circle, culminating in a Wavy Curve inequality.

The 'Lethal Collision' occurs when solving the algebraic inequality involving the area $S$ .

By substituting $k = x^2/y^2$ , the inequality transforms into $\frac{k - 4\sqrt{k} + 1}{(\sqrt{k} - 1)^2} \le 0$ .

Students routinely apply the method of intervals to the numerator but fail to account for the Denominator Restriction $(\sqrt{k} - 1)^2 \neq 0$ .

This oversight leads them to incorrectly include $k=1$ as a valid integer solution, resulting in the phantom sum of 91 instead of the correct 90.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Geometric Interpretation of Cross Product
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free