JEE Advanced · Mathematics · 3. Complex Numbers
Match the statements of Column I with these in Column II.
[Note : Here \(z\) takes values in the complex plane and \(\operatorname{Im}(z)\) and \(\operatorname{Re}(z)\) 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 \(4 / 5\) |
| (B) The set of points \(z\) satisfying \(|z+4|+|z-4|=0\) is contained in or equal to | (q) the set of points \(z\) satisfying \(\operatorname{Im}(z)=0\) |
| (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 \(|\operatorname{Im} z| \leq 1\) |
| (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 satisfying \(|\operatorname{Re} z| \leq 2\) |
| (t) the set of points \(z\) satisfying \(|z| \leq 3\) |
- A (A) q, (B) p, (C) p,s, (D) q,r,s
- B (A) q,r, (B) p, (C) p,s,t, (D) q,r,s,t
- C (A) q, (B) q, (C) p,s,t, (D) q,r,s,t
- D (A) q,r, (B) q, (C) p,s, (D) q,r,s
Answer & Solution
Correct Answer
(B) (A) q,r, (B) p, (C) p,s,t, (D) q,r,s,t
Step-by-step Solution
Detailed explanation
(A) \(z\) is equidistant from the points \(i|z|\) and \(-i|z|\), whose perpendicular bisector is \(\operatorname{Im}(z)=0\).
(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 \((\pm 4,0), a e=4\).
\(\therefore e=\frac{4}{5}\)
(C) \(|z| \leq|w|+\left|\frac{1}{w}\right|=\frac{5}{2} < 3\)
(D) \(|z| \leq|w|+\left|\frac{1}{w}\right|=2\)
\(\therefore \operatorname{Re}(z) \leq|z| \leq 2\)
(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 \((\pm 4,0), a e=4\).
\(\therefore e=\frac{4}{5}\)
(C) \(|z| \leq|w|+\left|\frac{1}{w}\right|=\frac{5}{2} < 3\)
(D) \(|z| \leq|w|+\left|\frac{1}{w}\right|=2\)
\(\therefore \operatorname{Re}(z) \leq|z| \leq 2\)
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