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Question #40867

Difficulty: 8/10 | 🟢 Free

The Question

The plane $\Pi: x - y + z - 9 = 0$ intersects the line passing through the points $P(1, 0, 2)$ and $Q(4, 3, 5)$ at the point $R$ . If $A(1, 2, 3)$ and $B(2, 4, 1)$ are two points in space, and $\Delta$ is the area of $\triangle RAB$ , find the value of $4\Delta^2$ .

Options

A.
45
B.
677
C.
17
D.
425

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Ratio of Division (k:1)|Area of Triangle via Determinants|Region Ambiguity

Why This Question Is Tricky

Common Mistakes

This question tests the application of the section formula for a line intersecting a plane, specifically targeting 'Region Ambiguity'.

The ratio $k$ is determined by $k = -\Pi(P)/\Pi(Q)$ .

A common trap occurs when students ignore the negative sign of the ratio (since both points yield negative values when substituted into the plane equation, meaning they lie on the same side of the plane).

If a student incorrectly assumes internal division ( $k=2$ ), they will calculate the erroneous intersection point $R(3, 2, 4)$ , which leads to the trap answer of 45.

Recognizing the external division is crucial for establishing the correct spatial coordinates before applying the determinant-based cross product for the triangle's area.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Ratio of Division (k:1)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free