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Question #40757

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and $S(3,0)$ be its focus. A variable point $P$ lies on $E$ . The point $Q$ divides the line segment $SP$ internally in the ratio $2:1$ (i.e., $SQ:QP = 2:1$ ). Let $C$ be the locus of $Q$ . If a line $L$ passing through the origin is tangent to $C$ , then the square of the slope of $L$ is:

Options

A.
$\frac{64}{91}$
B.
$\frac{16}{11}$
C.
$\frac{64}{109}$
D.
\text{No such real line exists}

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

Internal vs External Division|General Conic Definition|Condition of Tangency

Why This Question Is Tricky

Common Mistakes

This question tests the student's geometric intuition versus mechanical calculation.

The primary trap is blindly applying the condition of tangency ( $c^2 = a^2m^2 + b^2$ ) to find the slope of a tangent from a given point to a conic, without first verifying if the point lies outside the locus.

Here, the origin lies strictly inside the locus $C$ , making real tangents impossible.

Mechanical application yields a negative value for $m^2$ ( $-64/91$ ), which students often erroneously 'correct' to $64/91$ by dropping the negative sign.

Additionally, a secondary trap exists: if a student incorrectly applies the section formula ratio as $1:2$ , the origin falls outside that incorrect locus, yielding a real but incorrect slope squared of $16/11$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Internal vs External Division
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free