Finding Values of a & b in Geometry Problem

[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.

 

[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.

 

[tacman]

To find the values of a and b in this geometry problem, we can use the properties of perpendicular bisectors.

First, let's recall that a perpendicular bisector is a line that cuts another line segment into two equal parts at a right angle. This means that the line y = ax + 14 must pass through the midpoint of the line joining (1,2) and (b,6).

To find the midpoint of a line segment, we can use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

In this case, the midpoint would be ((1+b)/2, (2+6)/2) = ((1+b)/2, 4).

Now, since the line y = ax + 14 must pass through this midpoint, we can substitute these values into the equation:
4 = a((1+b)/2) + 14
4 = (a/2 + ab/2) + 14
-10 = a/2 + ab/2
-20 = a + ab

We now have an equation with two variables, a and b. To solve for these values, we need another equation.

Since the line y = ax + 14 is also the perpendicular bisector of the line joining (1,2) to (b,6), it means that the slope of the line joining these two points must be the negative reciprocal of the slope of the perpendicular bisector.

The slope of the line joining (1,2) and (b,6) can be found using the slope formula:
Slope = (y2 - y1)/(x2 - x1)
Slope = (6-2)/(b-1)
Slope = 4/(b-1)

The slope of the perpendicular bisector, on the other hand, is the negative reciprocal of this slope. Therefore, we can set up another equation:
a = -1/(4/(b-1))
a = -1*(b-1)/4
a = (-b+1)/4

Now we have two equations with two variables, a and b:
-20 = a + ab
a = (-b+1)/4

We can solve for one variable in terms of the other in one of the equations, and then substitute that into the other equation.

Let's solve for