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# 7.7: Apollonian Circle

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The exercises in this section are given as illustrations to the method of coordinates — it will not be used further in the sequel.

Exercise $$\PageIndex{1}$$

Show that for fixed real values $$a$$, $$b$$, and $$c$$ the equation

$$x^2 + y^2 + a \cdot x + b \cdot y + c = 0$$

describes a circle, one-point set or empty set.

Show that if it is a circle then it has center $$(- \dfrac{a}{2}, -\dfrac{b}{2})$$ and the radius $$r = \dfrac{1}{2} \cdot \sqrt{a^2 + b^2 - 4 \cdot c}$$.

Hint

Rewrite it the following way and think

$$(x + \dfrac{a}{2})^2 + (y + \dfrac{b}{2})^2 = (\dfrac{a}{2})^2 + (\dfrac{b}{2})^2 - c$$.

Exercise $$\PageIndex{2}$$

Use the previous exercise to show that given two distinct point $$A$$ and $$B$$ and positive real number $$k \ne 1$$, the locus of points $$M$$ such that $$AM = k \cdot BM$$ is a circle.

Hint

We can choose the coordinates so that $$B = (0, 0)$$ and $$A = (a, 0)$$ for some $$a > 0$$. If $$M = (x, y)$$, then the equation $$AM = k \cdot BM$$ can be written in coordinates as

$$k^2 \cdot (x^2 + y^2) = (x - a)^2 + y^2.$$

It remains to rewrite this equation as in Exercise $$\PageIndex{1}$$.

The circle in the exercise above is an example of the so-called Apollonian circle with focuses $$A$$ and $$B$$. Few of these circles for different values $$k$$ are shown on the diagram; for $$k = 1$$, it becomes the perpendicular bisector to $$[AB]$$.

Exercise $$\PageIndex{3}$$

Make a ruler-and-compass construction of an Apollonian circle with given focuses $$A$$ and $$B$$ thru a given point $$M$$.

Hint

Assume $$M \not\in (AB)$$. Show and use that the points $$P$$ and $$Q$$ constructed on the following diagram lie on the the Apollonian circle.