Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  OctNov  (P19709/12)  Q#5
Question
The line , where a and b are positive constants, intersects the x and yaxes at the points A and B respectively. The midpoint 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 yaxes at the points A and B respectively.
Let’s find the coordinates of the xintercept ie point A.
We have the equation of the line as;
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).
Hence, coordinates of .
Let’s now find the coordinates of the yintercept ie point B.
We have the equation of the line as;
The point at which curve (or line) intercepts yaxis, 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 midpoint of AB lies on the line .
Let’s first find the coordinates of midpoint of line AB with coordinates of point and found above.
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
Therefore;
xcoordinate of midpoint of the line
ycoordinate of midpoint of the line
Hence, coordinates of midpoint 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 yintercept of and hence cannot be zero. Therefore, is also not possible.
Hence;