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For $\alpha, \beta, z \in C$ and $\lambda>1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^{2}+|z-\beta|^{2}=2 \lambda$, then $|\alpha-\beta|$ is equal to \_\_\_\_.
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in: Jee Mains
- 1
- 1 week, 2 days ago
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Let C be the circle in the complex plane with centre $\mathrm{z}_{0}=\frac{1}{2}(1+3 \mathrm{i})$ and radius r = 1. Let $\mathrm{z}_{1}=1+\mathrm{i}$ and the complex number $\mathrm{z}_{2}$ be outside the circle C such that $\left|\mathrm{z}_{1}-\mathrm{z}_{0}\right|\left|\mathrm{z}_{2}-\mathrm{z}_{0}\right|=1$.
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- 1
- 1 week, 2 days ago
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Let $\mathrm{w}=\mathrm{z} \overline{\mathrm{z}}+\mathrm{k}_{1} \mathrm{z}+\mathrm{k}_{2} \mathrm{iz}+\lambda(1+\mathrm{i}), \mathrm{k}_{1}, \mathrm{k}_{2} \in \mathrm{R}$. Let $\mathrm{Re}(\mathrm{w})=0$ be the circle C of radius 1 in the first quadrant touching the line $\mathrm{y}=1$ and the $y$-axis. If the curve $\operatorname{Im}(w)=0$ intersects C at A and B, then $30(A B)^{2}$ is equal to \_\_\_\_.
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- 1
- 1 week, 2 days ago
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Let $\alpha=8-14 i, A=\left\{z \in C: \frac{a z-\bar{\alpha} \bar{z}}{z^{2}-(\bar{z})^{2}-112 i}=1\right\}$ and $B=\{z \in C:|z+3 i|=4\}$. The $\sum_{z \in A \cap B}(\operatorname{Rez}-\operatorname{Imz})$ is equal to \_\_\_\_.
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- 1
- 1 week, 2 days ago
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For all $\mathrm{z} \in \mathrm{C}$ on the curve $\mathrm{C}_{1}:|\mathrm{z}|=4$, let the locus of the point $\mathrm{z}+\frac{1}{\mathrm{z}}$ be the curve $\mathrm{C}_{2}$. Then
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- 1
- 1 week, 2 days ago
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Let $\mathrm{w}_{1}$ be the point obtained by the rotation of $\mathrm{z}_{1}=5+4 \mathrm{i}$ about the origin through a right angle in the anticlockwise direction,
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- 1
- 1 week, 2 days ago
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Let a, b be two real numbers such that $\mathrm{ab}<0$. If the complex number $\frac{1+\mathrm{ai}}{\mathrm{b}+\mathrm{i}}$ is of unit modulus and a + ib lies on the circle $|z-1|=|2 z|$, then a possible value of $\frac{1+[a]}{4 b}$, where [t] is greatest integer function, is
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- 1
- 1 week, 2 days ago
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Let $\mathrm{z}_{1}=2+3 \mathrm{i}$ and $\mathrm{z}_{2}=3+4 \mathrm{i}$. The set $\mathrm{S}=\left\{\mathrm{z} \in \mathrm{C}:\left|\mathrm{z}-\mathrm{z}_{1}\right|^{2}-\left|\mathrm{z}-\mathrm{z}_{2}\right|^{2}=\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}\right\}$ represents a
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- 1
- 1 week, 2 days ago
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Let z be a complex number such that $\left|\frac{z-2 i}{z+i}\right|=2, z \neq-i$. Then z lies on the circle of radius 2 and centre
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- 1
- 1 week, 2 days ago
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Let $\mathrm{S}=\{\mathrm{z} \in \mathrm{C},|\mathrm{z}-1|=1$ and $(\sqrt{2}-1)(\mathrm{z}+\overline{\mathrm{z}})-\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}})=2 \sqrt{2}\}$. Let $\mathrm{z}_{1}, \mathrm{z}_{2} \in \mathrm{~S}$ be such that $\left|\mathrm{z}_{1}\right|=\max _{\mathrm{Z} \in \mathrm{S}}|\mathrm{z}|$ and $\left|\mathrm{z}_{2}\right|=\min _{\mathrm{Z} \in \mathrm{S}}|\mathrm{z}|$. Then $\left|\sqrt{2} \mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}$ equals
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- 1
- 1 week, 2 days ago
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If the center and radius of the circle $\left|\frac{\mathrm{z}-2}{\mathrm{z}-3}\right|=2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3(\alpha, \beta, \gamma)$ is equal to
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- 1
- 1 week, 2 days ago
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Let z be a complex number such that the real part of $\frac{\mathrm{z}-2 \mathrm{i}}{\mathrm{z}+2 \mathrm{i}}$ is zero. Then, the maximum value of $|\mathrm{z}-(6+8 \mathrm{i})|$ is equal to
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- 1
- 1 week, 2 days ago
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Consider the following two statements : Statement I : For any two non-zero complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2},\left(\left|\mathrm{z}_{1}\right|+\left|\mathrm{z}_{2}\right|\right)\left|\frac{\mathrm{z}_{1}}{\left|\mathrm{z}_{1}\right|}+\frac{\mathrm{z}_{2}}{\left|\mathrm{z}_{2}\right|}\right| \leq 2\left(\left|\mathrm{z}_{1}\right|+\left|\mathrm{z}_{2}\right|\right)$,
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- 1
- 1 week, 2 days ago
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Let $\mathrm{S}_{1}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}| \leq 5\}, \mathrm{S}_{2}=\left\{\mathrm{z} \in \mathrm{C}: \operatorname{Im}\left(\frac{\mathrm{z}+1-\sqrt{3} \mathrm{i}}{1-\sqrt{3} \mathrm{i}}\right) \geq 0\right\}$ and $\mathrm{S}_{3}=\{\mathrm{z} \in \mathrm{C}: \operatorname{Re}(\mathrm{z}) \geq 0\}$. Then the area of the region $\mathrm{S}_{1} \cap \mathrm{~S}_{2} \cap \mathrm{~S}_{3}$ is
Started by:
- 1
- 1 week, 2 days ago
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For $\alpha, \beta, z \in C$ and $\lambda>1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^{2}+|z-\beta|^{2}=2 \lambda$, then $|\alpha-\beta|$ is equal to \_\_\_\_.
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