Question

If we plot $|Z_1-3-4i|=2$ and $|Z_2-3-4i|=5$, then on the argand plane, the locus of $Z_1$ and $Z_2$ is

• A. two circle touching each other
• B. two concentric circles
• C. two intersecting circles
• D. none of these

Answer 1

The correct option is B two concentric circles

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

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

center of circle$\left(1\right)=(h_1,k_1)=(3,4)$

center of circle $\left(2\right)=(h_2,k_2)=(3,4)$

Hence, circle have same center both center are cocentric.

Hence, two circle are cocentric circle.