Question

The inverse of the point z with respect to
the circle |z| = a is:​

Answer 1

Answer:

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Answer 2

Answer:

The inverse of a point z w.r.t. the circle |z| = a is $\frac{a^2}{\bar z}$.

Step-by-step explanation:

Let the inverse of point z be z'.

Given,

|z| = a

Calculation,

If z' is the inverse of a point z w.r.t. circle |z| = a then,

z, z' is collinear.

i.e.

arg(z) = arg(z')

As arg($\bar{z}$) = -arg(z)

Then, arg(z') = -arg($\bar{z}'$) = -arg($\bar{z}$)

⇒ arg(z') + arg($\bar{z}$) = 0

⇒ arg($z'$$\bar{z}$) = 0 (As arg($z_1$) + arg($z_2$) = arg($z_1z_2$))

⇒ $z'\bar z$ is real...(1) (If arg(z) = 0 then z is real)

Now we know that the point z and z’ lies on a circle with radius a, hence we can say.

|z| = a and |z'| = a

|z||z'| = a²

⇒ |$\bar z$||$z'$| = a²  (As $|\bar z| = |z|$)

But from equation (1)  $z'\bar z$ is real

$\implies \bar{z}z' =a^2$  (As |$\bar z$||$z'$| = a² and a² is real)

$\implies z' = \frac{a^2}{\bar z}$

Therefore, the inverse of a point z w.r.t. the circle |z| = a is $\frac{a^2}{\bar z}$.

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