Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01R)  Year 2013  June  Q#11
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Question
The line meets the the curve at the points A and B as shown in the figure.
a. Find the coordinates of A and the coordinates of B.
b. Find the distance AB in the form where r is a rational number.
Solution
a.
We are required to find the coordinates of the points of intersection of the given line and the curve.
If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the line and the curve).
Equation of the line is;
Equation of the curve is;
Substitute
Now we have two options.







Two values of x indicate that there are two intersection points.
With xcoordinate of point of intersection of two lines (or line and the curve) at hand, we can find the ycoordinate of the point of intersection of two lines (or line and the curve) by substituting value of xcoordinate of the point of intersection in any of the two equations.
We choose equation of the line;
For 
For 






Hence, the line and the curve intersect at points with coordinates
It is evident from the figure that point A is on the positive side of xaxis and B on the negative side of xaxis, therefore, A
b.
We are required o find the distance AB.
Expression for the distance between two given points
Therefore, we need coordinates of both points A and B which we have already found in (a) as A
Hence;
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