Question

Let M be the point (2, 3) and let l be a line through M which meets 2x + y −3 = 0 at A and meets 3x −2y + 1 = 0 at B. If M is the midpoint of AB, find the equation of l.

Answer 1

Let A = (a,b) and B = (c,d). The midpoint M of AB is (2, 3) = ((a + c)/2, (b + d)/2), so

a +c = 4  ....(1) 

b +d = 6. ....(2) 

Since the point A(a,b) lies on the line 2x + y −3 = 0 and the point B(c,d) lies on the line 3x − 2y + 1 = 0, we also have 

2a + b = 3 .....(3) 

3c −2d = −1. ....(4)

From (1) and (2), we have c = 4− a and d = 6−b. 

Substituting these into (4) gives 

3a − 2b = 1. (5) 

We can now solve (3) and (5) to obtain a = 1 and b = 1. 

We have found that A is (a,b) = (1, 1). So the gradient of the line l is m = (1−3)/(1−2) = 2. Thus l has equation y −1 = 2(x −1) or, equivalently, y = 2x −1.