Question

Suppose A(a,-2a+4) where a < -2 and B(b, 2b+4) where b>-2, and O(2, 0). Show that m(A, O)=-m(O, B).

Answer 1

Slope of AO=(2a-4)/(2-a)=2(a-2)/(2-a)=-2 and slope of OB=-(2b+4)/(2-b)=2(b+2)/(b-2).

The constraints on a and b are that a,b≠2. So a<-2 is OK but b>-2 is insufficient.

Slope of AB=2(a+b)/(b-a).

If m(A,O)=-m(O,B) then 2(b+2)/(b-2)=2.

2b+4=2b-4 which leads to 4=-4 which is false. Therefore there is a mistake in the question.

I believe that the location of B should be B(b,-2b+4) so the slope of OB is 2(b-2)/(b-2)=2.

Then it’s clear that slope of AO=-2 and OB=2 so m(A,O)=-m(O,B).