Locus / Geometry Question (1 Viewer)

[Jase]

Let P(x,y) be a point and L_1 and L_2 be two lines in the number plane. Let C be the set of all points P such that the sum of the squares of the distances of P from L_1 and L_2 is r^2.
Prove that C is a circle if and only if L_1 and L_2 are perpendicular.

 

[Sober]

Let P(x,y) be a point and L_1 and L_2 be two lines in the number plane. Let C be the set of all points P such that the sum of the squares of the distances of P from L_1 and L_2 is r^2.
Prove that C is a circle if and only if L_1 and L_2 are perpendicular.

Since the only important factor about the two lines is the angle they form at the intersection then you can restrict then to:

L_1: y=0
L_2: y=mx

Let f(x,y) be the perpindicular distance from L_2:

y<SUP>2</SUP> + f(x,y)<SUP>2</SUP> = r<SUP>2</SUP>

y<SUP>2</SUP> + f(x,y)<SUP>2</SUP> = x<SUP>2</SUP> + y<SUP>2</SUP>

f(x,y) = |x|

This shows that L_2 is the y-axis therefore the two lines are perpendicular.