Finding Values of a & b in Geometry Problem

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The discussion centers on finding the values of a and b for the equation y = ax + 14, which serves as the perpendicular bisector of the line segment connecting points (1,2) and (b,6). The user initially struggled but eventually derived two equations: b = 1 - 4a and b = -1 - (20/a). By solving these equations simultaneously, the user successfully determined the values of a and b. They sought feedback on the method used and whether there was a more straightforward approach. The solution process highlights the relationship between the gradients of perpendicular lines and the use of midpoints in geometry.
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"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 :confused: 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.
 
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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.
 
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