Find the Locus Equation: Distance Ratio of Points P, Q, and R

Solve for the equation of the locus of point R given the fixed points P(2,5) and Q(4,7) and the distance relationship 5|PR| = 3|QR|.

Answer

Let the coordinates of point

, point

, and point

.
We are given the condition

.
Squaring both sides of the equation, we get

, which simplifies to

The distance formula between two points

and

.
Therefore,

.
And

Substitute these into the squared equation:

Expand the squared terms:

Distribute the coefficients:

Combine like terms on each side:

Move all terms to one side to form the equation of the locus:

Simplify the terms:

Divide the entire equation by 4 to simplify:

This is the equation of the locus of

The final answer is

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Find the Locus Equation: Distance Ratio of Points P, Q, and R

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