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Question #40503

Difficulty: 8/10 | 🟢 Free

The Question

Let $E_1$ be the quadratic equation $x^2 + ax + b = 0$ , where $a, b \in \mathbb{R}$ and $b \neq 0$ . The point $P(a, b)$ is the pole of the chord of contact $L$ with respect to the parabola $x^2 = 4y$ . Let $L$ intersect the x-axis and y-axis at points $A(\alpha, 0)$ and $B(0, \beta)$ respectively. If the quadratic equation $E_2$ whose roots are $\alpha$ and $\beta$ has at least one common root with $E_1$ , then the locus of $P$ consists of a line and a parabola. If the line intersects the parabola at points $M$ and $N$ , find the length of the line segment $MN$ .

Options

A.
$\frac{8\sqrt{2}}{3}$
B.
$\frac{16\sqrt{2}}{3}$
C.
$\frac{4\sqrt{5}}{3}$
D.
$\frac{8\sqrt{5}}{3}$

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

Condition for Common Roots|Chord of Contact Logic|Polar and Pole Duality

Why This Question Is Tricky

Common Mistakes

This question fuses coordinate geometry (pole and polar duality) with the algebraic theory of equations (condition for common roots).

The primary trap lies in the interpretation of the 'common root' condition: students often attempt to use the standard resultant method $(c_1a_2 - c_2a_1)^2 = (a_1b_2 - a_2b_1)(b_1c_2 - b_2c_1)$ by explicitly forming $E_2$ , which leads to a massive, error-prone degree-4 algebraic mess.

The conceptual bypass is recognizing that the roots of $E_2$ are explicitly known (the intercepts $\alpha$ and $\beta$ ), so the common root condition simply means substituting these intercepts directly into $E_1$ .

A secondary trap is failing to reject $b=0$ , which would lead to a degenerate conic or trivial roots, testing the student's rigor in handling discriminant and domain constraints.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Condition for Common Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free