# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#6

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 .