Question

The points P and Q have coordinates (–1, 6) and (9, 0) respectively.

The line  is perpendicular to PQ and passes through the mid-point of PQ.

Find an equation for , giving your answer in the form ax + by + c = 0, where a, b and c are integers.

Solution

We are required to find equation of line .

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We have neither coordinates of a point on line  nor slope of the line .

First we find coordinates of a point on line .

We are given the line  passes through the mid-point of the line PQ and is also perpendicular to PQ. 

We first find coordinates of the mid-point of line PQ.

We are given that points P and Q have coordinates (–1, 6) and (9, 0) respectively. 

To find the mid-point of a line we must have the coordinates of the end-points of the line.

Expressions for coordinates of mid-point of a line joining points  and;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Therefore;

x-coordinate of mid-point  of the PQ

y-coordinate of mid-point  of the PQ

Hence, coordinates of mid-point of line PQ are (4,3).

Next, we find slope of line .

We are given the line  is perpendicular to PQ.

If a line  is normal to the curve , then product of their slopes  and  at that point (where line  is normal to the curve) is;

Therefore;

Hence, if we have slope of the line PQ we can find slope of the line .

We can find slope of line PQ as follows.

Expression for slope (gradient) of a line joining points  and ;

We are given that points P and Q have coordinates (–1, 6) and (9, 0) respectively.

Therefore;

Hence, gradient of the line ;

With coordinates of a point M (4,3) on the line  and slope of the line  at hand , we can write equation of this line.

Point-Slope form of the equation of the line is;