Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#4

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Question

The equation of a curve is  and the equation of a line is , where  is a constant.

    i.       In the case where , find the coordinates of the points of intersection of the line
and the curve.

   ii.       Find the value of  for which the line is a tangent to the curve.

Solution

    i.
 

For the case where , the equation of line becomes;

Equation of the curve is given as;

We are required to find the coordinates of the points of intersection of 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;

We solve this equation as follows;

Now we have two options;

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

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

We choose equation of the line;

Therefore;

For

For

Hence point of intersection of  and .


ii.
 

If a given line is tangent to the curve then the given line intersects the curve only at a single point.

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).

Rearranged equation of the curve from (i);

Equation of the line is;

Equating both equations;

For a quadratic equation , the expression for solution is;

Where  is called discriminant.

If , the equation will have two roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

Therefore if the line and the curve intersect at only single point then;

For the given equation;

Hence;

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