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Question #71664

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be the auxiliary circle of the ellipse $E: \frac{x^2}{16} + \frac{y^2}{9} = 1$ in the $xy$ -plane. A tangent line $T$ is drawn to $C$ at a point $P$ in the first quadrant such that the area of the triangle formed by $T$ and the coordinate axes is $16$ . Let $Q$ be the point on $E$ corresponding to $P$ , and let $\Pi$ be the tangent plane to the elliptical cylinder $\frac{x^2}{16} + \frac{y^2}{9} = 1$ along the generator passing through $Q$ . A line $L$ passing through $S(0,0,\sqrt{2})$ is given by the parametric equations $x = 3r$ , $y = 4r$ , $z = \sqrt{2} + 12r$ . If $L$ intersects $\Pi$ at a point $R$ , what is the exact distance $SR$ ?

Options

A.
$\frac{156\sqrt{2}}{25}$
B.
$\frac{12\sqrt{2}}{25}$
C.
$\frac{144\sqrt{2}}{25}$
D.
$\frac{169\sqrt{2}}{25}$

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

Line-Plane intersection logic|Parametric form manipulation|Parametric form pitfalls|Ellipse Auxiliary Circle

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 integrates 3D geometry with conic sections, specifically testing the intersection of a parametrically defined line with a tangent plane to an elliptical cylinder.

The core trap lies in the parametric representation of the line L.

Students frequently assume that the parameter 'r' in (x0 + lr, y0 + mr, z0 + nr) directly represents the Euclidean distance from the base point.

However, this is only true if the direction vector is normalized (i.e., l^2+m^2+n^2=1).

By setting the direction ratios to <3, 4, 12>, the parameter 'r' must be scaled by the magnitude of the vector (13) to find the true distance.

The preliminary step involving the auxiliary circle ensures a multi-layered problem where an error in the initial eccentric angle calculation or the final distance metric leads to incorrect options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line-Plane intersection logic
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free