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Question #40772

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real values of $m$ for which the line $y = m\left(x - \frac{3}{2}\right)$ is tangent to the curve defined by $\sqrt{(x-3)^2 + y^2} = 2(x - 2)$ . Find the number of elements in $S$ .

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

Family of Lines|General Conic Definition|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize extraneous solutions introduced by squaring an equation involving radicals, specifically within the context of conic sections.

The equation $\sqrt{(x-3)^2 + y^2} = 2(x-2)$ represents only the right branch of a hyperbola due to the implicit domain restriction $x \ge 2$ .

When students substitute the family of lines and set the discriminant to zero, they find two real slopes.

However, these slopes correspond to tangents touching the left branch of the squared hyperbola at $x = -1$ .

Failing to verify the point of tangency against the original geometric constraint leads to the incorrect conclusion that two valid tangents exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Family of Lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free