Let A = {z in mathbb{C} : |z - 2| le 4} and B | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The set $A$ represents a disk centered at $2 + 0i$ with radius $R = 4$. The boundary is $|z - 2| = 4$.The set $B$ represents an ellipse with foci

Vedclass · Accepted Answer

$\frac{17}{2}$

Vedclass · Answer

(C) The set $A$ represents a disk centered at $2 + 0i$ with radius $R = 4$. The boundary is $|z - 2| = 4$.The set $B$ represents an ellipse with foci at $2$ and $-2$. The sum of distances from the foci is $2a = 5$, so $a = \frac{5}{2}$. The center is at the origin $(0, 0)$.The equation of the ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a = \frac{5}{2}$ and $c = 2$. Since $b^2 = a^2 - c^2$, we have $b^2 = \frac{25}{4} - 4 = \frac{9}{4}$, so $b = \frac{3}{2}$.The ellipse is $\frac{x^2}{(5/2)^2} + \frac{y^2}{(3/2)^2} = 1$.To maximize $|z_1 - z_2|$, we choose $z_1$ on the boundary of $A$ and $z_2$ on the ellipse $B$ such that they are as far apart as possible.The point on the ellipse $B$ furthest from the center $2$ is $z_2 = -\frac{5}{2}$.The point on the circle $A$ furthest from $z_2 = -\frac{5}{2}$ is $z_1 = 6$.Thus, the maximum distance is $|6 - (-5/2)| = |6 + 2.5| = 8.5 = \frac{17}{2}$.

Let $A = \{z \in \mathbb{C} : |z - 2| \le 4\}$ and $B = \{z \in \mathbb{C} : |z - 2| + |z + 2| = 5\}$. Then the maximum value of $\{|z_1 - z_2| : z_1 \in A \text{ and } z_2 \in B\}$ is

Similar Questions

If $|z-3 i|+|z+5 i|=4$,then the locus of $z$ is

If $P(x, y)$ denotes $z = x + iy$ in the Argand plane and $\left|\frac{z-1}{z+2i}\right| = 1$,then the locus of $P$ is a/an

If ${\tan ^{ - 1}}(\alpha + i\beta ) = x + iy,$ then $x =$

The figure in the complex plane given by $10 z \bar{z} - 3(z^2 + \bar{z}^2) + 4i(z^2 - \bar{z}^2) = 0$ is

For any integer $k$,let $w_k = \cos \left( \frac{k\pi}{11} \right) + i \sin \left( \frac{k\pi}{11} \right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^8 |w_{2k+1} - w_{2k}|}{\sum_{k=1}^4 |w_{3k-1} - w_{3k-2}|}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module