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Question #41350

Difficulty: 8/10 | 🟢 Free

The Question

A plane is tangent to the surface $S: z = 4 - x^2 - y^2$ at a point $P(x_0, y_0, z_0)$ in the first octant ( $x_0 > 0, y_0 > 0, z_0 > 0$ ). The tangent plane intersects the coordinate axes at points $A(a, 0, 0)$ , $B(0, b, 0)$ , and $C(0, 0, c)$ . If $x_0 = y_0$ and the vector $\vec{v} = \frac{1}{a}\hat{i} + \frac{1}{b}\hat{j} + \frac{1}{c}\hat{k}$ has a magnitude of $\frac{1}{2}$ , what is the volume of the tetrahedron $OABC$ (where $O$ is the origin)?

Options

A.
$9$
B.
$18$
C.
$27$
D.
$54$

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

Tangent/Normal Slope Interplay|Variable plane intercepts|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between a surface's gradient vector and the geometric intercepts of its tangent plane.

A common trap is for students to incorrectly equate the gradient vector $\nabla F$ directly to the intercept vector $\langle 1/a, 1/b, 1/c \rangle$ , failing to normalize the plane equation by the constant term $W = \nabla F \cdot \vec{r}_0$ .

This omission invalidates the magnitude relationship and leads to unsolvable equations.

Furthermore, the problem requires students to geometrically interpret the magnitude of the intercept vector as the inverse of the perpendicular distance to the origin, and carefully apply octant constraints to eliminate extraneous roots.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Tangent/Normal Slope Interplay
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free