# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#8

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**Question**

Points A and B have coordinates (h, h) and (4h + 6, 5h) respectively. The equation of the perpendicular bisector of AB is 3x + 2y = k. Find the values of the constants h and k.

**Solution**

We are given that line AB has coordinates of the two points and .

We are also given that equation of perpendicular bisector of AB is;

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

Therefore, we can substitute slopes of line AB and its perpendicular bisector in above equation.

Therefore, first we find slope of line AB with points and as follows.

Expression for slope of a line joining points and ;

Hence;

Similarly, next we find slope of perpendicular bisector of line AB with equation as follows.

Slope-Intercept form of the equation of the line;

Where is the slope of the line.

Now, we can rewrite the given equation of perpendicular bisector of line AB as;

Hence, slope of perpendicular bisector of line AB is;

We can now substitute slopes of line AB and its perpendicular bisector in following

equation.

Since we are given equation of the perpendicular bisector of line AB that means mid- point of AB also lies on its perpendicular bisector.

We know that if a point lies on a curve/line then it must satisfy equation of that curve/line.

Therefore, first we find the coordinates of mid-point of line AB and then substitute these coordinates in equation of the perpendicular bisector of line AB to find the value of .

To find the mid-point of a line we must have the coordinates of the end-points of the line.

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

We are given that line AB has coordinates of the two points and .

We have found above that . Therefore, coordinates of the two points are and .

Hence, coordinates of the mid-point of the line AB can be found as follows;

x-coordinate of mid-point of the line

y-coordinate of mid-point of the line

Therefore, coordinates of the mid-point of the line AB are (8, 6) and since this point also lies on perpendicular bisector of line AB, these coordinates must satisfy equation of the perpendicular bisector of line AB.

Equation of perpendicular bisector of line AB is given as;

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