Locus and compound locus?

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The Question:

The locus of points that are 5 units from A is a circle whose center is A.

The locus of points that are equidistant from both A and B is the perpendicular bisector of line AB.

The intersection of these two loci answers your question.

The equation of the circle whose center is A and whose radius is 5 units, if the origin (0,0) is at A, is

x^2 +y^2 = 5^2

x^2 +y^2 = 25 -----------first locus.

If A again is the origin (0,0), the perpendicular bisector of line AB is

x = 6/2

x = 3 ---------------------second locus

Their intersection:

Substitute 3 for x in the first locus,

3^2 +y^2 = 25

y^2 = 25 -9 = 16

y = +4 or -4

So there are two intersection points, (3,4) and (3,-4)

Therefore, two points. -----------answer.