Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2017  MayJun  (P19709/13)  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 xcoordinates of point P.
To find the corresponding possible ycoordinates of of point P, we can utilize the equation obtained in (i);
For 
For 






Hence, possible coordinates of point P are;


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