Circle - The Constant Ratio Curve

There are many definitions of a circle. There may also be many ways of generating a circle. A circle is simply the locus of all points equidistant from a point, which is the center. However, there is also another way to define a circle, and it is simply remarkable. After investigating the constant product curve, the lemniscate, I decided to see what the curve for a constant ratio would look like. The constant ratio curve is the result of this.

The locus of all points such that the ratio of the distances between the point in the locus and two fixed points is a circle, with the condition that the ratio does not equal 1.

In other words, let the two points be A₁(–a,0) and A₂(a,0), and let the point P(x, y) be in the locus. Now, let the distance between A₁ and P be d₁ and let distance between A₂ and P be d₂. If d₁/d₂ = k for some constant k, then the points P lie on a circle. The image below shows our setup.

Derivation of Circle Equation

Deriving the equation of this curve is simple enough. From the image above, we can derive the distance d₁ and d₂ as follows:

(i) $d_{1}^{2} = (x+a)^2+y^2$

(ii) $d_{2}^{2} = (x-a)^2+y^2$

Since the distances are in a ratio k, then we have:

(iii) $k^2 = \frac{(x+a)^2+y^2}{(x-a)^2+y^2}$

We can manipulate equation (iii) to give us the equation of a circle:

(iv) $\left( x + \frac{1+k^2}{1-k^2} \right )^2 + y^2 = \left (\frac{2ak}{1-k^2}\right )^2$

Thus, the equation of the circle has a radius of $r = \left|\frac{2ak}{1-k^2}\right|$, for k ≠ 1.

When k = 1, the equation of the locus P is x = 0. This is the equation of a line that passes through the midpoint of a segment that connects A₁ and A₂ and is perpendicular to the segment. This can be visualized intuitively. The image below shows our curve - actually two curves that give the ratio of k.

These two circles are congruent. These circles are positioned symmetric to the midpoint of A₁ and A₂. Note that the order in which we take the ratio of the distances matters. If we take the ratio d₁/d₂ = k, then this represents one circle. The other circle with ratio of –k represents the ratio d₂/d₁ = 1/k.

Also, notice that the center of the circle lies on the segment or a line continuous with the segment passing through A₁ and A₂. As k approaches 1, the radius of the two circles approaches infinity—or that the size of the two circles increases. The two circles converge to form the line x = 0. This is because the ratio is approaching 1.

As k approaches 0, then the radius of the two circles approaches 0 and their centers get closer and closer to the two points A₁ and A₂. When k = 0, the locus P represents the two points A₁ and A₂.

You can interact with the Geogebra activity below to play around with how a and k affect the circles.