Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  OctNov  (P19709/13)  Q#6
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Question
Three points, A, B and C, are such that B is the midpoint 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 midpoint between points A(2,m) and C.
Expressions for coordinates of midpoint of a line joining points and;
xcoordinate of midpoint of the line
ycoordinate of midpoint of the line
Therefore;
xcoordinate of midpoint of the line
ycoordinate of midpoint of the line
Substituting all given coordinates;












Hence coordinates of point C are (2n2, m12).
ii.
We are given equation of a line as;
This line passes through point C and is perpendicular to line AB.
SlopeIntercept 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 (2n2, m12).
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 .
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