Question

$|Z_1-3-4i|=2$ and $|Z_2-3-4i|=5$, then the maximum distance between the $Z_1$ and $Z_2$ is

• A. $5$
• B. $7$
• C. $3$
• D. $12$

Answer 1

The correct option is B $7$

Given,

$|z_1-3-4i|=2$ and $|z_2-3-4i|=5$

To find the locus of given equations.

Solution,

Let$z_1=x_1+{iy}_1\&z_2=x_2+{iy}_2$

Put the value of $z_1\&z_2$ in the given equation.

$|x_1+{iy}_1-3-4i|=2$ $|x_2+{iy}_2-3-4i|=5$

$|x_1-3+{i(y}_1-4)|=2$ $|x_2-3+{i(y}_2-4)|=5$

By solving modulus we get. By solving modulus we get

$\left|{\sqrt{{(x_1-3)}^2+{(y}_1-4)}}^2\right|=2$ $\left|{\sqrt{{(x_2-3)}^2+{(y}_2-4)}}^2\right|=5$

Squaring both sides we get. Squaring both sides we get

${{(x_1-3)}^2+{(y}_1-4)}^2=4⟶\left(1\right)$ ${{(x_2-3)}^2+{(y}_2-4)}^2=25⟶\left(2\right)$

On comparing with stander equation of circle

${{(x-h)}^2+(y-k)}^2=r^2$ Where,$(h,k)$ represent center of circle.

From equation $\left(1\right)$

$r_1=2$

similarly equation$\left(2\right).$

$r_2=5$
Where $r_1\&r_2$ are the radius of a circles.

for maximum distance b/w $z_1\&z_2$ is $r_1+r_2$

maximum distance$=r_1+r_2=5+2=7$

Hence the answer is $7$.