If the mid-point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0 find the value of k.
Solution
We have two points A (3, 4) and B (k, 7) such that its mid-point is P(x,y).
It is also given that point P lies on a line whose equation is
2x + 2y+ 1 = 0
In general to find the mid-point P(x,y) of two points `A(x_1,y_1)` and `B(x_2, y_2)` we use section formula as
`P(x,y) = ((x_1+x_2)/2,(y_1+y_2)/2)`
Therefore mid-point P of side AB can be written as
`P(x,y) = ((k + 3)/2, (7 + 4)/2)`
Now equate the individual terms to get,
`x= (k + 3)/2`
`y = 11/2`
Since, P lies on the given line. So,
2x + 2y + 1 = 0
Put the values of co-ordinates of point P in the equation of line to get,
`2((k + 3))/2+2(11/2) + 1 = 0`
On further simplification we get,
k + 15 = 0
So, k = -15
