Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2004  MayJun  (P19709/01)  Q#6
Question
The curve and the line intersect at two points. Find
i. the coordinates of the two points,
ii. the equation of the perpendicular bisector of the line joining the two points.
Solution
i.
To find the coordinates of intersection points;
If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the line and the
curve).
Equation of the line is;
Equation of the curve is;
Equating both equations;
Solution of Quadratic equation;
In the given case;
We have two options now;






Two values of x indicate that there are two intersection points. Corresponding values of y coordinate can be found by substituting values of x in any of the two equation i.e either equation of the line or equation of the curve.
We choose equation of the line;
For ; 
For ; 








Hence the two points of intersection are and .
ii.
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).
We are required to write the equation of the perpendicular bisector of the line joining the two points and , therefore we first find the point on the perpendicular bisector and that is the midpoint of line joining points and and then find the slope of the perpendicular bisector.
To find the midpoint on the perpendicular bisector;
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
For the given case;
xcoordinate of midpoint of
the line
ycoordinate of midpoint of
the line
Therefore the midpoint of the line joining points and is;
To find the slope of the perpendicular bisector;
If two lines are perpendicular (normal) to each other, then product of their slopes and is;
Now, to find the slope of the line joining points and ;
Expression for slope of a line joining points and ;
Therefore, slope of the line joining points and ;
Hence, slope of the perpendicular bisector;
Now we can write the equation of the perpendicular bisector with point and slope ;
PointSlope form of the equation of the line is;
Therefore,
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