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Question #41742

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ be the line of intersection of the planes $P_1: x + y + z = 2$ and $P_2: x - y + 2z = 3$ . Let $L_2$ be the line of intersection of the planes $P_3: 2x + y - z = 1$ and $P_4: \alpha x + \beta y + z = 4$ . It is given that $L_1$ and $L_2$ are coplanar distinct lines. If the plane containing $L_1$ and $L_2$ is perpendicular to the plane $3x - 5y + z = 0$ , find the value of $\alpha^2 + \beta^2$ .

Options

A.
10
B.
13
C.
25
D.
29

Check Your Answer

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

Symmetric vs Non-symmetric form|Geometric Interpretation of Cross Product|Singular vs Non-Singular Matrices|System of Linear Equations (Cramer/Inverse)

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of systems of linear equations and the cross product.

The primary trap lies in the non-symmetric representation of lines.

Students often attempt to convert these to symmetric forms, which is algebraically heavy and prone to errors.

The elegant approach recognizes that coplanarity of lines defined by four planes implies the $4 \times 4$ augmented matrix is singular (or the $3 \times 3$ system yields a point satisfying the 4th equation).

Furthermore, it embeds a secondary singular matrix trap: the perpendicularity condition translates to a zero scalar triple product $[\vec{d}_1, \vec{d}_2, \vec{n}_{perp}] = 0$ , forcing students to evaluate a determinant involving unknown variables derived from cross products.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Symmetric vs Non-symmetric form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free