Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2015  OctNov  (P19709/12)  Q#6
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Question
Points A, B and C have coordinates A(3,7), B(5,1) and C(1,k), where k is a constant.
i. Given that AB=BC, calculate the possible values of k.
The perpendicular bisector of AB intersects the xaxis at D.
ii. Calculate the coordinates of D.
Solution
i.
Expression to find distance between two given points and is:
We are given that A(3,7), B(5,1) and C(1,k) and;
Now we have two options.




ii.
We are required to find coordinates of D which is the xintercept of the perpendicular bisector of AB.
For this we need equation of the perpendicular bisector of AB.
To find the equation of the line either we need coordinates of the two points on the line (TwoPoint form of Equation of Line) or coordinates of one point on the line and slope of the line (PointSlope form of Equation of Line).
First we find coordinates of a point on the perpendicular bisector of AB.
The common point between AB and its perpendicular bisector is the midpoint of AB where perpendicular bisector of any line intersects the line.
Let’s find midpoint of AB. We already have a A(3,7), B(5,1).
To find the midpoint of a line we must have the coordinates of the endpoints of the line.
Expressions for coordinates of midpoint of a line joining points and;
xcoordinate of midpoint of the line
ycoordinate of midpoint of the line
xcoordinate of midpoint of the line
ycoordinate of midpoint of the line
Now we have coordinates of a point on the perpendicular bisector of AB i.e. M(1,4).
Next we need slope of the perpendicular bisector of AB to write its equation.
If two lines are perpendicular (normal) to each other, then product of their slopes and is;
Therefore, if we can find slope of AB we can find slope of its perpendicular bisector.
Expression for slope of a line joining points and ;
From A(3,7), B(5,1) we can find slope of AB.
We know that;
Hence;
Now we can write equation of the perpendicular bisector of AB.
Coordinates of a point on perpendicular bisector of AB and its slope are (1,4) and .
PointSlope form of the equation of the line is;
This is equation of the perpendicular bisector of AB. Now we can find coordinates of its xintercept D.
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, coordinates of point D (2,0).
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