The number of values of z C, satisfying the equations |z-(4+8i)|=√10 and |z-(3+5i)|+|z-(5+…

Question

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

Accepted Answer

The second equation describes an ellipse with foci $S_1(3,5)$ and $S_2(5,11)$. Here $2ae=|S_1S_2|=\sqrt{(5-3)^2+(11-5)^2}=\sqrt{4+36}=2\sqrt{10}$, and $2a=4\sqrt5\Rightarrow a=2\sqrt5$. So $ae=\sqrt{10}\Rightarrow e=\dfrac{1}{\sqrt2}$, and $b^2=a^2(1-e^2)=20\left(\dfrac12\right)=10$.

The centre of the ellipse (midpoint of the foci) is $\left(\dfrac{3+5}2,\dfrac{5+11}2\right)=(4,8)$.

The first equation is a circle centred at $(4,8)$ with radius $\sqrt{10}=\sqrt{b^2}=b$ --- exactly the semi-minor axis length. So this circle is concentric with the ellipse and its radius equals the ellipse's semi-minor axis, meaning the circle touches the ellipse precisely at the two endpoints of the minor axis.

Hence there are exactly 2 common points.

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