Exploring the Locus of Points with Given Distance Ratios: The Apollonius Circle

In geometry, the concept of the locus of points satisfying a given condition is fundamental. One interesting geometric locus is the set of points O such that the ratio of the distances from O to two fixed points A and B is a constant. This locus is known as the Apollonius Circle. In this article, we delve into the mathematical derivation of such a locus, focusing on the specific case when OA/OB 1/2.

Mathematical Derivation

Let A and B be fixed points in the plane, with coordinates A(x1, y1) and B(x2, y2), respectively. We aim to find the locus of point O(x, y) such that the ratio OA/OB k, where k 1/2.

Step 1: Setting Up the Ratio

The given condition can be expressed as:

[ frac{OA}{OB} frac{k}{1} ]

Step 2: Using the Distance Formula

Using the distance formula, we get:

[ frac{sqrt{(x - x_1)^2 (y - y_1)^2}}{sqrt{(x - x_2)^2 (y - y_2)^2}} frac{1}{2} ]

Step 3: Squaring Both Sides

Squaring both sides to eliminate the square roots:

[ left(frac{sqrt{(x - x_1)^2 (y - y_1)^2}}{sqrt{(x - x_2)^2 (y - y_2)^2}}right)^2 left(frac{1}{2}right)^2 ][ frac{(x - x_1)^2 (y - y_1)^2}{(x - x_2)^2 (y - y_2)^2} frac{1}{4} ]

Step 4: Cross-Multiplying and Rearranging

Cross-multiplying and simplifying:

[ 4[(x - x_1)^2 (y - y_1)^2] (x - x_2)^2 (y - y_2)^2 ][ 4(x^2 - 2x_1x x_1^2 y^2 - 2y_1y y_1^2) x^2 - 2x_2x x_2^2 y^2 - 2y_2y y_2^2 ][ 4x^2 - 8x_1x 4x_1^2 4y^2 - 8y_1y 4y_1^2 x^2 - 2x_2x x_2^2 y^2 - 2y_2y y_2^2 ]

Step 5: Combining Like Terms and Rearranging

Combining and rearranging the terms to obtain a standard form:

[ 4x^2 - 4y^2 - 2x_2x 2x_1x - 8y_1y 8y_2y 4x_1^2 - 8y_1^2 x_2^2 - y_2^2 0 ][ 3x^2 3y^2 2x_1x - 2x_2x - 8y_1y 8y_2y 4x_1^2 - 4y_1^2 x_2^2 - y_2^2 0 ]

This results in the equation of a circle if the coefficients are properly arranged.

Geometric Insight

The locus of points O forming a given ratio with two fixed points A and B is a circle known as the Apollonius Circle. In our specific case where the ratio is 1/2, the center and radius of the Apollonius Circle can be calculated from the equation derived.

General Case

For any given ratio k, the locus of points O satisfying OA/OB k is also a circle, known as the Apollonius Circle. The coordinates and the shape of this circle are determined by the specific ratio and the coordinates of points A and B.

Conclusion

The Apollonius Circle provides a fascinating insight into the relationship between distances and geometric loci. By understanding the steps and the mathematical derivation, one can explore various scenarios and applications of such geometric loci in both pure and applied mathematics.