Match the statements in Column I with those in Column II. [Note: Here takes values in the complex plane and and denote, respectively, the imaginary part and the real part of .] [2010]

Question

Match the statements in Column I with those in Column II. [2010]

[Note: Here $z$ takes values in the complex plane and $Im z$ and $Re z$ denote, respectively, the imaginary part and the real part of $z$ .]

	Column I		Column II
(A)	The set of points $z$ satisfying $\| z - i \| z \| \| = \| z + i \| z \| \|$ is contained in or equal to	(p)	an ellipse with eccentricity $\frac{4}{5}$
(B)	The set of points $z$ satisfying $\| z + 4 \| + \| z - 4 \| = 10$ is contained in or equal to	(q)	the set of points $z$ satisfying $Im z = 0$
(C)	If $\| w \| = 2$ , then the set of points $z = w - \frac{1}{w}$ is contained in or equal to	(r)	the set of points $z$ satisfying $\| Im z \| \leq 1$
(D)	If $\| w \| = 1$ , then the set of points $z = w + \frac{1}{w}$ is contained in or equal to	(s)	the set of points $z$ satisfying $\| Re z \| < 2$
		(t)	the set of points $z$ satisfying $\| z \| \leq 3$

Smart Achievers · Accepted Answer

(1)

$(A) \to (q, r)$

$Re z$

$\Rightarrow z is equidistant from two points (0, | z |) and (0, - | z |), which lie on imaginary axis.$

$∴$ $z must lie on real axis \Rightarrow Im (z) = 0 . Also | I_{m} (z) | \leq 1$

$(B) \to p$

$Sum of distances of z from two points (- 4, 0) and (4, 0) is 10 which is greater than 8 .$

$∴$ $z traces an ellipse with 2 a = 10 and 2 a e = 8$

$\Rightarrow e = \frac{4}{5}$

$(C) \to (p, s, t)$

$Let ω = 2 (\cos θ + i \sin θ), then$

$z = ω - \frac{1}{ω} = 2 (\cos θ + i \sin θ) - \frac{1}{2} (\cos θ - i \sin θ)$

$\Rightarrow x + i y = \frac{3}{2} \cos θ + i \frac{5}{2} \sin θ$

$Here,$ $| z |$ $= \sqrt{\frac{9 + 25}{4}} = \sqrt{\frac{34}{4}} \leq 3 and$ $| R_{e} (z) |$ $\leq 2$

$Also x = \frac{3}{2} \cos θ, y = \frac{5}{2} \sin θ \Rightarrow \frac{4 x^{2}}{9} + \frac{4 y^{2}}{25} = 1$

$Which is an ellipse with e = \sqrt{1 - \frac{9}{25}} = \frac{4}{5}$

$(D) \to (q, r, s, t)$

$Let ω = \cos θ + i \sin θ then z = 2 \cos θ \Rightarrow Im (z) = 0$

$Also$ $| z | \leq 3 and | Im (z) | \leq 1, | R_{e} (z) | \leq 2$

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