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 twice the distance from $(0,3)$

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Key Concepts

Distance Formula

The Distance Formula is a fundamental tool in coordinate geometry that calculates the distance between two points in the Cartesian plane using the formula ?((x2 - x1)² + (y2 - y1)²). It is essential for converting geometric distance problems into algebraic equations that can be manipulated and solved.

Locus of Points

A locus of points is a set of all points that satisfy a certain condition or a group of conditions. In coordinate geometry, determining a locus often involves establishing an equation that represents all points fulfilling a specific distance or ratio requirement relative to fixed points or lines.

Apollonius Circle

An Apollonius circle is defined as the set of points in the plane that have a constant ratio of distances to two fixed points. When the distance from one fixed point is a constant multiple of the distance from another fixed point, the resulting locus is a circle, unless the ratio equals one, in which case the locus is the perpendicular bisector of the segment joining the two points.

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