Question

Three points, A, B and C, are such that B is the mid-point of AC. The coordinates of A are (2,m) and  the coordinates of B are (n,-6), where m and n are constants.

i.       Find the coordinates of C in terms of m and n.

The line y =x + 1 passes through C and is perpendicular to AB.

   ii.       Find the values of m and n.

Solution

     i.
 

We are given that point B(n,-6) is mid-point between points A(2,m) and C.

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 line

y-coordinate of mid-point  of the line

Substituting all given coordinates;

Hence coordinates of point C are (2n-2, -m-12).

   ii.
 

We are given equation of a line as;

This line passes through point C and is perpendicular to line AB.

Slope-Intercept form of the equation of the line;

Where  is the slope of the line.

Therefore slope of the given line is;

We are also given that this line is perpendicular to line AB.

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

Therefore;

We can find slope of the line AB.

Expression for slope of a line joining points  and ;

Therefore for points A(2,m) and B(n,-6).

Hence;

We are also given that line passes through point C, therefore, equation of the line must satisfy the  coordinates of point C,

We have found in (i) that coordinates of point C are (2n-2, -m-12).

We can substitute these coordinates in the equation of the line;

Substituting above found value of ;

We substitute this value of in the expression of ;

Hence,  and .