# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#5

Question

The line , where a and b are positive constants, intersects the x- and y-axes at the points A  and B respectively. The mid-point of AB lies on the line  and the distance .  Find the values of a and b.

Solution

We need to work through the problem statement very carefully to glean the information scattered  therein.

We are given that line   intersects the x- and y-axes at the points A and B respectively.

Let’s find the coordinates of the x-intercept ie point A.

We have the equation of the line as;

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).

Hence, coordinates of .

Let’s now find the coordinates of the y-intercept ie point B.

We have the equation of the line as;

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Hence, coordinates of .

We are given that the mid-point of AB lies on the line .

Let’s first find the coordinates of mid-point of line AB with coordinates of point  and  found above.

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

Therefore;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Hence, coordinates of mid-point of AB are .

Since  lies on the line with equation , coordinates of M must satisfy the

equation of this line.

We are also given that .

Expression to find distance between two given points  and is:

We have coordinates of point  and  found above. Hence;

We have found above that , therefore, substituting in above equation;

Now we have two options.

We have the equation;

 For ; For ;

It is evident that  is not possible because  is y-intercept of  and hence  cannot  be zero. Therefore,  is also not possible.

Hence;