# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#8

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Question

A(-1,1) and P(a,b) are two points, where a and b are constants. The gradient of AP is 2.

(i)        Find an expression for b in terms of a.

(ii)      B(10,-1) is a third point such that AP = AB. Calculate the coordinates of the possible positions           of P.

Solution

(i)

We are given coordinates of two points A(-1,1) and P(a,b).

We are also given gradient of line AP as; Expression for slope of a line joining points and ; Therefore, for AP;        (ii)

We are given that; We will find both AB and AP and equate to find the coordinates of point P.

We are given coordinates of all points A(-1,1) and P(a,b) and B(10,-1).

As we have found in (i) therefore coordinates of point P(a,2a+3)

Expression to find distance between two given points and is:  Since ;                  Now we have two options.      We have found two possible x-coordinates of point P.

To find the corresponding possible y-coordinates of of point P, we can utilize the equation obtained  in (i); For For       Hence, possible coordinates of point P are;  