# 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

Hits: 348

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;  