Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | 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 x-axis 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 x-intercept 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 (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  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 mid-point of AB where perpendicular bisector of any line intersects the line.

Let’s find mid-point of AB. We already have a A(-3,7), B(5,1).

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

x-coordinate of mid-point  of the line

y-coordinate of mid-point  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 .

Point-Slope form of the equation of the line is;

This is equation of the perpendicular bisector of AB. Now we can find coordinates of its x-intercept  D.

The point  at which curve (or line) intercepts x-axis, 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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