# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#3

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Question

The diagram shows the curve  and the line  intersecting at points A, O and B.

i.       Show that the x-coordinates of A and B satisfy the equation .

ii.       Solve the equation  and hence find the coordinates of A and B, giving your answers in an exact form.

Solution

i.

We are given that points A & B are the intersection points 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;

Equating both equations;

ii.

Let;

Hence;

Now we have two options.

 Since Impossible

Hence we have;

Two values of x indicate that there are two intersection points.

Corresponding values of y coordinate can be found by substituting values of x in any of the two equation i.e either equation of the line or equation of the curve.

We choose;

Therefore;

 For For

Hence  and