Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2006  MayJun  (P19709/01)  Q#5
Question
The curve intersects the line at two points. Find the distance between the two points.
Solution
Expression to find distance between two given points and is:
So first we need to find the coordinates of points of intersection of the curve and the line.
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).
In this case, equation of the curve is;
We can rewrite it as;
Equation of the line is;
We can rewrite it as;
Equating both equations;
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 the equation of line;
We can rewrite it as;












Hence the two points of intersection of the curve and the line are and .
Now we can find the distance between these two points;
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