Question
The number of values of $z \in \mathbb{C}$, satisfying the equations
$|z - (4 + 8i)| = \sqrt{10}$ and $|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$, is :
The correct answer is
2
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.
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