# Geometry problem

1. Jan 14, 2006

### m00c0w

"Find all possible values of a and b given that y = ax + 14 is the perpendicular bisector of the line joining (1,2) to (b,6)" I'm totally stuck I've tried all the methods I could think of but they all lead to dead ends. I'm hoping someone can point me in the right direction.
Many thanks.

2. Jan 14, 2006

### m00c0w

Ok guys I managed to solve it :D I was wondering if you could me whether my method was the best to use in the situation though and whether or not there was a simpler way. Here goes:

A(1,2) B(b,6)

I worked out the gradient of AB to be 4/(b-1)

I knew that the product of the gradient of AB and the line y = ax + 14 had to be -1 as they are perpendicular, so 4/(b-1) had to be -1/a

I rearranged that to get b = 1 - 4a

I worked out the coordinates of the mid-point of AB to be ((1+b)/2 , 4) so I then substitued that into y = ax + 14 and rearranged to get b = -1-(20/a)

So...

#1 b = 1 - 4a
#2 b = -1 - (20/a)

I then solved them simultaneously to get the values of a and then substitued those into equation #1 to get the values of b.