$|z - (4 + 8i)| = \sqrt{10}$ and $|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$, is :
We are looking for the number of complex values '$z$' that satisfy two equations simultaneously.
Equation 1: Circle
The first equation is $|z - (4 + 8i)| = \sqrt{10}$.
- This represents a circle in the complex plane.
- Center: $C = 4 + 8i$.
- Radius: $r = \sqrt{10}$.
Equation 2: Ellipse
The second equation is $|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$.
- This represents an ellipse in the complex plane.
- The foci are $F_1 = 3 + 5i$ and $F_2 = 5 + 11i$.
- The sum of the distances from any point on the ellipse to the foci is $2a = 4\sqrt{5}$, which means the semi-major axis length is $a = 2\sqrt{5}$.
Ellipse Parameters Calculation
We calculate the distance between the foci ($2c$) and the center of the ellipse:
- Distance between foci: $2c = |F_2 - F_1| = |(5 + 11i) - (3 + 5i)| = |2 + 6i|$ $2c = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}$. Therefore, $c = \sqrt{10}$.
- Center of the ellipse: $C = \frac{F_1 + F_2}{2} = \frac{(3 + 5i) + (5 + 11i)}{2} = \frac{8 + 16i}{2} = 4 + 8i$.
- Calculate the semi-minor axis length ($b$) using the relation $a^2 = b^2 + c^2$: $a^2 = (2\sqrt{5})^2 = 20$. $c^2 = (\sqrt{10})^2 = 10$. $b^2 = a^2 - c^2 = 20 - 10 = 10$. $b = \sqrt{10}$.
Comparing Circle and Ellipse
Now, we compare the parameters of the circle and the ellipse:
- Both the circle and the ellipse share the same center $C = 4 + 8i$.
- Circle radius $r = \sqrt{10}$.
- Ellipse semi-major axis $a = 2\sqrt{5} = \sqrt{20}$.
- Ellipse semi-minor axis $b = \sqrt{10}$.
Determining Intersection Points
The key observation is that the circle's radius is equal to the ellipse's semi-minor axis length ($r = b = \sqrt{10}$).
- The distance from the center $C$ to the endpoints of the minor axis (co-vertices) is $b = \sqrt{10}$. Since this equals the circle's radius $r$, the endpoints of the minor axis lie on the circle.
- The distance from the center $C$ to the endpoints of the major axis (vertices) is $a = \sqrt{20}$. Since $a > r$, the vertices lie outside the circle.
- Since the circle and ellipse share the same center, and the circle's radius matches the minimum distance ($b$) from the center to points on the ellipse, the circle intersects the ellipse exactly at the two endpoints of the minor axis.
Therefore, there are 2 values of $z$ satisfying both equations.
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
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If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
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Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
