The number of values of z C, satisfying the equations and , is : [2026]

Question

The number of values of z $\in$ C, satisfying the equations $| z - (4 + 8 i) | = \sqrt{10}$ and $| z - (3 + 5 i) | + | z - (5 + 11 i) | = 4 \sqrt{5}$ , is : [2026]

Smart Achievers · Accepted Answer

(2)

$\in$

It is a circle with center (4, 8) and radius = $\sqrt{10}$

$| z - (3 + 5 i) | + | z - (5 + 11 i) | = 4 \sqrt{5}$

It is an ellipse with focii S(3, 5) and S'(5, 11)

$S S' = \sqrt{4 + 36} = \sqrt{40} = 2 \sqrt{10}$

Clearly $4 \sqrt{5} > S S'$

Center of ellipse $(\frac{3 + 5}{2}, \frac{5 + 11}{2}) \equiv (4, 8)$ coincides with center of circle.

Now, $2 a = 4 \sqrt{5} \Rightarrow a = 2 \sqrt{5}$

$b^{2} = a^{2} - c^{2} \Rightarrow b^{2} = 20 - 10 = 10$

$\Rightarrow b = \sqrt{10}$ = radius of circle

$∴$ Ellipse and circle will intersect at end points of minor axis.

So, 2 points of intersection.

Solution

Q.
The number of values of z $\in$ C, satisfying the equations $| z - (4 + 8 i) | = \sqrt{10}$ and $| z - (3 + 5 i) | + | z - (5 + 11 i) | = 4 \sqrt{5}$ , is : [2026]

1 0

2 2

3 1

4 4

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Solution

Q. The number of values of z ∈ C, satisfying the equations |z–(4+8i)|=10 and |z–(3+5i)|+|z–(5+11i)|=45, is : [2026]

1 0

2 2

3 1

4 4

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Q.
The number of values of z $\in$ C, satisfying the equations $| z - (4 + 8 i) | = \sqrt{10}$ and $| z - (3 + 5 i) | + | z - (5 + 11 i) | = 4 \sqrt{5}$ , is : [2026]