In Exercises 87 and 88, find an equation of the graph that...

In Exercises 87 and $88,$ find an equation of the graph that consists of all points $(x, y)$ having the given distance from the origin. (For a review of the Distance Formula, see Appendix C.)

The distance from the origin is $K(K \neq 1)$ times the distance from $(2,0) .$

Key Concepts

Distance Formula

The distance formula is a fundamental tool in analytic geometry used to calculate the distance between two points in the coordinate plane. It is derived from the Pythagorean theorem and provides the algebraic means to express distances, which is essential when setting up equations based on geometric conditions.

Locus of Points

A locus in geometry is the set of all points that satisfy a specific condition or a set of conditions. In problems where a relationship between distances is specified, establishing the locus often involves deriving an equation that every point meeting that condition must satisfy.

Apollonius Circle

An Apollonius circle is defined as the set of all points in the plane for which the ratio of distances to two fixed points is constant. This concept is significant when the distance from one fixed point is a constant multiple of the distance from another. The resulting curve is usually a circle, derived by equating the distances through algebraic manipulation.

Equation Derivation from Distance Conditions

Deriving an equation from given distance conditions involves expressing the distances using the distance formula and then equating them based on the specified relationship. This method is a cornerstone in translating geometric conditions into algebraic equations, making it easier to analyze and graph the resulting figure.

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