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Question #40797

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $t$ such that the vertex of the parabola $P_t: y = x^2 - 2tx + t^6 - 21t^4 + 85t^2 - 64$ lies on the $x$ -axis, and the line $L: y = -2\\sqrt{39}x - 39$ does not intersect $P_t$ . Find the value of $\\sum_{t \\in S} t$ .

Options

A.
7
B.
0
C.
-7
D.
21

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Parabola Shifted Vertex|Point-Line Relative Position|Polynomial Function Parity|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

Students often solve the high-parity polynomial $t^6 - 21t^4 + 84t^2 - 64 = 0$ to find symmetric roots $t \\in \\{\\pm 1, \\pm 2, \\pm 4\\}$ .

A common trap is to assume all these roots are valid and sum them to 0, or to sum their squares to 21, ignoring the geometric constraint.

The Extraneous Solution Rejection trap requires applying the discriminant condition from the point-line relative position, which breaks the parity and strictly eliminates the negative roots, leaving only the positive solutions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Parabola Shifted Vertex
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free