Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#10

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Question

A straight line has equation , where  is a constant, and a curve has equation .

     i.       Show that the x-coordinates of any points of intersection of the line and curve are given by the equation .

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

  iii.        The two tangents, given by the values of  found in part (ii), touch the curve at points A and B. Find the coordinates of A and B and the equation of the line AB.

Solution


i.
 

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;

Hence, we can find x-coordinates of points of intersection of line and the curve from this equation.


ii.
 

Line will be tangent to the curve if it intersects the curve at a single point.

We can solve the equation obtained in (i);

We can see that it is a quadratic equation.

Standard form of quadratic equation is;

Expression for discriminant of a quadratic equation is;

If   ;        Quadratic equation has two real roots.

If   ;        Quadratic equation has no real roots.

If   ;       Quadratic equation has one real root/two equal roots.

Since line is tangent to the curve it intersects the curve at a single point and therefore, this equation will have one real root or two equal/repeated roots.

Hence, for this equation;

For the given equation;

Therefore;

Again we have a quadratic equation to solve for the values of . But this time we can
factorize the equation rather than solving through quadratic formula.

Now we have two options.


iii.
 

From (ii), two values of  indicate that there are two intersection points and therefore line makes two tangents to the curve.

We again have two options for the coordinates of points of intersection of the line and the curve.

For

For

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 equation of line;

 is obtained through

𝑦=
 equation of the line
when
.

 is obtained through

𝑦=
 equation of the line
when
.

Therefore, two points of intersection (or where line is tangent to the curve)

Hence we have  and .

Now we are required to find the equation of the line AB.

Two-Point form of the equation of the line is;

For the given case we can write it as;

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