If P is a moving point in the rectangular coordinate plane such that the distances between P and the two points (−7, 16) and (−14, 11) are always equal, then the equation of the locus of P is

Question

A. 7x + 5y + 6 = 0.

B. 5x + 7y = 0.

C. 7x − 5y + 141 = 0.

D. 5x − 7y + 147 = 0.

Answer 1

The answer is A. Here’s why.

Join the points and treat this as the base of an isosceles triangle. The perpendicular bisector of the base is the locus of P. So let’s find the midpoint of the join: ((-14-7)/2,(11+16)/2)=(-21/2,27/2).

Now find the slope of the join: (16-11)/(-7-(-14))=5/7. The slope of the perpendicular is -7/5.

The equation of the perpendicular can be found by plugging in the midpoint coordinates:

y-27/2=-7(x+21/2)/5, double this:

2y-27=-7(2x+21)/5, multiply through by 5:

10y-135=-14x-147,

14x+10y+12=0=7x+5y+6=0.