Question

The equation $\frac{\left|z-5i\right|}{\left|z+5i\right|}=12$, where $z=x+iy$, represents a/an

• A. Circle
• B. Ellipse
• C. Parabola
• D. None of these

Answer 1

The correct option is A

Circle

Explanation of the correct option:

Simplify the given equation.

The equation $\frac{\left|z-5i\right|}{\left|z+5i\right|}=12$ is given.

Simplify the above equation as follows.

$$\begin{gathered} \left|z-5i\right|=12\left|z+5i\right| \\ \Rightarrow \left|x+iy-5i\right|=12\left|x+iy+5i\right| \\ \Rightarrow \left|x+i\left(y-5\right)\right|=12\left|x+i\left(y+5\right)\right| \\ \Rightarrow \sqrt{x^2+{(y-5)}^2}=12\sqrt{x^2+{(y+5)}^2} \end{gathered}$$

Take square on both sides.

$$\begin{gathered} x^2+{(y-5)}^2=144(x^2+{(y+5)}^2) \\ \Rightarrow x^2+y^2-10y+25=144(x^2+y^2+10y+25) \\ \Rightarrow x^2+y^2-10y+25=144x^2+144y^2+1440y+3600 \\ \Rightarrow 143x^2+143y^2+1450y+3575=0 \end{gathered}$$

Which is the equation of a circle.

Therefore, the given equation represents a circle.

Hence, option $A$is the correct .