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Question #41638

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_t$ be the family of quadric surfaces defined by $(t^2 - t)x^2 + (t-1)y^2 + (z-3)^2 = t-1$ , where $t \in \mathbb{R}$ . Let $\Pi_t$ be the polar plane of the point $P(0, 0, 5)$ with respect to $S_t$ . Let $C_t$ be the intersection of $\Pi_t$ and $S_t$ . Let $Z$ be the set of all $z$ -coordinates of the points where $C_t$ intersects the $z$ -axis, considering all $t \in \mathbb{R}$ for which such an intersection exists. Find the sum of the elements of $Z$ .

Options

A.
4
B.
5
C.
6
D.
8

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

Parametric form pitfalls|Chord of Contact Logic|The Zero-Factor Cancellation Trap|Implicit to Explicit Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the 'Zero-Factor Cancellation Trap' within the context of 3D parametric quadric surfaces.

Students commonly simplify the initial surface equation by dividing by $(t-1)$ , transforming it into $tx^2 + y^2 + \frac{(z-3)^2}{t-1} = 1$ .

This implicit-to-explicit conversion prematurely assumes $t \neq 1$ , causing them to lose the degenerate quadric surface (a double plane) that provides a valid geometric intersection.

By missing $t=1$ , they only find $t=5$ (yielding $z=5$ ) and select the trap option, failing to account for the singular chord of contact at $z=3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Parametric form pitfalls
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free