The locus of points $z$ which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$, is a circle

Question

Use algebra to prove that the locus of points z which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$ and $c = a + bi$ any fixed complex number, is a circle centre $O$. Give the radius of the circle in terms of $k$ and $|c|$.

I squared both sides and got this: $$(k^2−1)x^2+(k^2−1)y^2+(a^2+b^2-k^2a^2-k^2b^2)k^2=0$$

I might have gone wrong somewhere though.

Edit. Never mind, I didn't go wrong.

$$(k^2-1)x^2+(k^2-1)y^2-(k^2-1)k^2a^2-(k^2-1)k^2b^2=0$$ $$x^2+y^2=k^2(a^2+b^2)$$ $$r^2=k^2(a^2+b^2)$$ $$r=k|c|$$

Answer 1

Hint: If you square both sides, and expand out (initially writing things in terms of your variables and their complex conjugates), you will get a lot of useful cancellation.

Here is the beginning of such a calculation, to help you out.

Squaring the left hand side yields:

$$(z-k^2c)(\overline{z}-k^2\overline{c})=|z|^2+k^4|c|^2-k^2(z\overline{c}+\overline{z}c).$$

Similarly, squaring the right hand side yields:

$$k^2(|z|^2+|c|^2-(z\overline{c}+\overline{z}c)).$$

Setting the two sides equal and rearranging, we have

$$(k^2-1)|z|^2=(k^4-k^2)|c|^2.$$

Answer 2

instead of squaring let me see if conjugates help.

$\begin{eqnarray} 0 = |z - k^2c|^2 -k^2 |z -c|^2 &=& (z - k^2 c)(\bar z - k^2 \bar c) -k^2(z-c)(\bar z - \bar c) \\ &=& z \bar z - k^2 c \bar z - k^2\bar c z + k^4 c \bar c -k^2(z \bar z - c \bar z -z \bar c + c \bar c) )\\ &=&(1-k^2)z \bar z - k^2(1-k^2)c\bar c \\ &=&(1-k^2)(|z|^2 - k^2|c|^2) \end{eqnarray}$

$ \mbox{ if } k^2 \neq 1, \mbox{ then } |z|^2 = k^2|c|^2$

Answer 3

Here are the steps $$ \left|z - k^2c\right| = k\left|z - c\right| $$ $$ \left|z - k^2c\right|^2 = k^2\left|z - c\right|^2 $$ $$ \left(z - k^2c\right)\left(\overline{z - k^2c}\right) = k^2(z - c)\left(\overline{z-c}\right) $$ $$ \left(z - k^2c\right)\left(\overline{z} - k^2\overline{c}\right) = k^2(z - c)\left(\overline{z}-\overline{c}\right) $$ $$ \left|z\right|^2 - k^2\overline{c}z-k^2c\overline{z}+k^4\left|c\right|^2= k^2\left(\left|z\right|^2-\overline{c}z-c\overline{z} + \left|c\right|^2\right) $$ $$ \left|z\right|^2 - k^2\overline{c}z-k^2c\overline{z}+k^4\left|c\right|^2= k^2\left|z\right|^2-k^2\overline{c}z-k^2c\overline{z} + k^2\left|c\right|^2 $$ $$ \left|z\right|^2 -k^2c\overline{z}+k^4\left|c\right|^2= k^2\left|z\right|^2-k^2c\overline{z} + k^2\left|c\right|^2 $$ $$ \left|z\right|^2 +k^4\left|c\right|^2= k^2\left|z\right|^2 + k^2\left|c\right|^2 $$ $$ \left|z\right|^2 -k^2\left|z\right|^2= k^2\left|c\right|^2 -k^4\left|c\right|^2$$ $$ \left|z\right|^2\left(1 -k^2\right)= k^2\left|c\right|^2\left(1 -k^2\right)$$ $$ \left|z\right|^2= k^2\left|c\right|^2$$ $$ \left|z\right|= k\left|c\right|$$