JEE Advance - Mathematics (2010 - Paper 2 Offline - No. 2)
Match the statements in Column I with those in Column II.
(A) The set of points z satisfying $$\left| {z - i} \right|\left. {z\,} \right\|\,\, = \left| {z + i} \right|\left. {\,z} \right\|$$ is contained in or equal to
(B) The set of points z satisfying $$\left| {z + 4} \right| + \,\left| {z - 4} \right| = 10$$ is contained in or equal to
(C) If $$\left| w \right|$$= 2, then the set of points $$z = w - {1 \over w}$$ is contained in or equal to
(D) If $$\left| w \right|$$ = 1, then the set of points $$z = w + {1 \over w}$$ is contained in or equal to.
(p) an ellipse with eccentricity $${4 \over 5}$$
(q) the set of points z satisfying Im z = 0
(r) the set of points z satisfying $$\left| {{\rm{Im }}\,{\rm{z }}} \right| \le 1$$
(s) the set of points z satisfying $$\,\left| {{\mathop{\rm Re}\nolimits} \,\,z} \right| < 2$$
(t) the set of points z satisfying $$\left| {\,z} \right| \le 3$$
[Note : Here z takes value in the complex plane and Im z and Re z denotes, respectively, the imaginary part and the real part of z.]
Column I
(A) The set of points z satisfying $$\left| {z - i} \right|\left. {z\,} \right\|\,\, = \left| {z + i} \right|\left. {\,z} \right\|$$ is contained in or equal to
(B) The set of points z satisfying $$\left| {z + 4} \right| + \,\left| {z - 4} \right| = 10$$ is contained in or equal to
(C) If $$\left| w \right|$$= 2, then the set of points $$z = w - {1 \over w}$$ is contained in or equal to
(D) If $$\left| w \right|$$ = 1, then the set of points $$z = w + {1 \over w}$$ is contained in or equal to.
Column II
(p) an ellipse with eccentricity $${4 \over 5}$$
(q) the set of points z satisfying Im z = 0
(r) the set of points z satisfying $$\left| {{\rm{Im }}\,{\rm{z }}} \right| \le 1$$
(s) the set of points z satisfying $$\,\left| {{\mathop{\rm Re}\nolimits} \,\,z} \right| < 2$$
(t) the set of points z satisfying $$\left| {\,z} \right| \le 3$$
(A) - q, s ; (B) - p ; (C) - p, t ; (D) - q, r, s, t
(A) - q, r ; (B) - p ; (C) - p, s, t ; (D) - q, r, s, t
(A) - p, r ; (B) - p ; (C) - p, t ; (D) -q, r, s, t
(A) - p ; (B) - q ; (C) - r, s ; (D) -q, r, s, t
Explanation
(A) z is equidistant from the points $$i|z|$$ and $$ - i|z|$$, whose perpendicular bisector is $${\mathop{\rm Im}\nolimits} (z) = 0$$.
(B) Sum of distance of z from (4, 0) and ($$-$$4, 0) is a constant 10, hence locus of z is ellipse with semi-major axis 5 and focus at ($$\pm$$ 4, 0), ae = 4.
$$\therefore$$ $$e = {4 \over 5}$$
(C) $$|z| \le |w| + \left| {{1 \over w}} \right| = {5 \over 2} < 3$$
(D) $$|z| \le |w| + \left| {{1 \over w}} \right| = 2$$
$$\therefore$$ $${\mathop{\rm Re}\nolimits} (z) \le |z| \le 2$$
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