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Question #40892

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be the origin. The position vectors of the vertices $B$ and $C$ of $\triangle ABC$ are $\vec{b} = 12\hat{i} + 4\hat{j} + 3\hat{k}$ and $\vec{c} = 4\hat{i} + 3\hat{j}$ respectively. Let $D$ and $E$ be the points where the internal and external angle bisectors of $\angle A$ intersect the line $BC$ , respectively. What is the area of $\triangle ADE$ ?

Options

A.
$\frac{1625}{144}$
B.
$\frac{5}{13}$
C.
25
D.
$\frac{325}{36}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Section Formula Internally/Externally|Geometric Interpretation of Cross Product|Angle Bisector Direction

Why This Question Is Tricky

Common Mistakes

This question tests the geometric application of the section formula via the Angle Bisector Theorem and the properties of the cross product.

The 'Lethal Collision' occurs if students assume the direction vectors of the bisectors are simply $\vec{b} + \vec{c}$ and $\vec{b} - \vec{c}$ without normalizing them, leading to an incorrect area of 25.

Another trap is using the normalized vectors $\hat{b} + \hat{c}$ and $\hat{b} - \hat{c}$ directly as the position vectors of D and E, yielding 5/13.

The correct approach requires mapping the ratio of the magnitudes (13:5) to the section formula to find the exact position vectors on the line BC before computing the vector area.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Section Formula Internally/Externally
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free