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Question #41630

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_1: 2x + y + 3z = 6$ and $P_2: x + 3y + 2z = 4$ be two planes. Let $\Pi$ be a plane passing through the line of intersection of $P_1$ and $P_2$ , and bisecting one of the angles between them. The normal vector $\vec{n} = a\hat{i} + b\hat{j} + c\hat{k}$ to the plane $\Pi$ is chosen such that $a, b, c$ are coprime integers with $a > 0$ . It is given that the projection of $\vec{n}$ onto the $xy$ -plane is a vector whose tip lies on the director circle of the ellipse $16x^2 + 9y^2 = 144$ (assuming its tail is at the origin). If $\Pi$ intersects the coordinate axes at points $A, B,$ and $C$ , then the area of $\triangle ABC$ is:

Options

A.
$\frac{25\sqrt{2}}{6}$
B.
$\sqrt{6}$
C.
$\frac{25\sqrt{3}}{6}$
D.
$2\sqrt{6}$

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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

Equation of plane through intersection|Angle Bisector Direction|Director Circle Locus|Angle Bisector Logic

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to synthesize 3D geometry (angle bisectors of planes) with 2D coordinate geometry (director circle of an ellipse).

The primary trap lies in the selection of the correct bisector.

Students often memorize the acute/obtuse angle bisector conditions ( $a_1a_2+b_1b_2+c_1c_2 \gtrless 0$ ) and might blindly apply them, assuming the question implies one of these standard geometric properties.

By introducing the director circle locus as a projection constraint on the normal vector, the problem forces an analytical verification.

Furthermore, confusing the director circle ( $x^2+y^2=a^2+b^2$ ) with the auxiliary circle ( $x^2+y^2=\max(a^2,b^2)$ ) will lead to a dead end, acting as a secondary conceptual trap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Equation of plane through intersection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free