Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#6

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Question

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

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

   ii.       For the case where the line intersects the curve at two points, it is given that the x-coordinate  of one of the points of intersection is -1. Find the x-coordinate of the other point of intersection. 

  iii.       For the case where the line is a tangent to the curve at a point P, find the value of  and the coordinates of P.

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;

   ii.
 

We can find the coordinates of both points of intersection of the given curve and the line from  equation found in (i) if we have the value of ;

It is given that the x-coordinate of one of the points of intersection is -1. 

With x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the  y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value  of x-coordinate of the point of intersection in any of the two equations.

We choose equation of the curve;

Substituting
;

Hence one of the points of intersection of the given curve and the line is . We can use these  coordinates to find value of .

Substituting  and  in;

Now we can write  as;

Now we have two options.

X-coordinate of second point of intersection is 5.

  iii.
 

When line is tangent to the curve then there is only one point of intersection of line and the curve.

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.

From (i), we have;

When line is tangent to the curve, solution of this quadratic equation must be single i.e repeated  roots. 

Therefore;

Hence, we can write the equation   as;

Therefore;

As expected we have found repeated roots when line is tangent to the curve.

With x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the  y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value  of x-coordinate of the point of intersection in any of the two equations.

We choose equation of the curve;

Hence, coordinates of point .

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