Question

The point A has coordinates (-2,5) and the point B has coordinates (8,-6).

a.   Find and equation for the straight line AB, giving your answer in the form , where p, q and r are integers.

b.   The point C has coordinates (k,k+1). Given that the angle ACB is a right angle, find the two  possible values of k.

Solution

a.
 

We are required to find equation of a line AB.

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 are given coordinates of both points A(-2,5) and B(8,-6).

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

Therefore for line AB;

b.    

We are given that angle ACB is a right angle which means lines AC and BC are perpendicular. 

If two lines are perpendicular (normal) to each other, then product of their slopes  and  is;

Therefore;

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

Therefore, slopes of lines AC and BC are;

Hence;

We are given coordinates of points A(-2,5), B(8,-6) and C(k,k+1). Therefore;

Now we have two options.