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Question #40806

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ . Let $S$ be the set of all complex numbers $z = x+iy$ lying on the director circle of $E$ . For each $z \in S$ , let $m_1$ and $m_2$ be the slopes of the two tangents drawn from $z$ to $E$ , ordered such that $m_1 > m_2$ . Define a complex number $\omega = (y - m_1 x) + i(y - m_2 x)$ . If $z_0 \in S$ satisfies $\arg(\omega) = \frac{3\pi}{4}$ , find the value of $z_0$ .

Options

A.
5
B.
-5
C.
5i
D.
-5i

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

Director Circle Locus|Euler's Form and Rotations|The Quadrant Adjustment Logic

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle the principal branch of the argument function when mapping algebraic locus conditions to the complex plane.

A common trap is to blindly square the condition $\arg(\omega) = 3\pi/4$ into $\tan \theta = -1$ , leading to the equation $y - m_1 x = -(y - m_2 x)$ .

Solving this yields two candidate points, $z = 5$ and $z = -5$ .

Students who fail to perform the 'Quadrant Adjustment' check—verifying that $\text{Re}(\omega) < 0$ and $\text{Im}(\omega) > 0$ —will incorrectly assume both points are valid or guess the wrong sign, missing the discontinuity of the Arg function that restricts the locus to a single point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Director Circle Locus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free