Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#9

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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  and the curve.

   ii.       Find the set of values of  for which  does not intersect the curve.

  iii.       In the case where , one of the points of intersection is P (2, 6). Find the angle, in degrees correct to 1 decimal place, between  and the tangent to the curve at P.

Solution


i.
 

To find the coordinates of the points of intersection;

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 a line  becomes;

It can be rewritten as;

However, equation of a curve is;

It can be rewritten as;

Therefore;

Now we have two options;

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

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 two points of intersection of line and the curve are  and .


ii.
 

To find the coordinates of the point of intersection;

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 a line  becomes;

It can be rewritten as;

However, equation of a curve is;

It can be rewritten as;

Therefore;

If the line and the curve do not intersect then above equation does not have any solution/root.

Solution of Quadratic equation;

In the given case;

To determine the number of the roots of the quadratic equation we can utilize the discriminant;

If   ; there are two roots.

If   ; there is only one root.

If   ; there are no roots.

In this case;

Therefore;

So line and the curve do not intersect for;


iii.
 

To find the angle between line  and tangent to the curve at point , first we need angles of both line  and tangent to the curve at point P from x-axis.

First we find the angle of line  from x-axis;

In this case equation of a line  becomes;

It can be rewritten as;

Slope-Intercept form of the equation of the line;

Where  is the slope of the line.

Hence slope of the line ;

Therefore;

Now we need to find slope of the tangent to the curve at point ;

The slope of a curve  at a particular point is equal to the slope of the tangent to the curve at the same point;

So we need to fins the slope of the curve at point ;

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that particular point.

Gradient (slope)  of the curve  at a particular point  can be found by substituting x-coordinates of that point in the expression for gradient of the curve;

In the given case:

We can rewrite the equation as;

Rule for differentiation of  is:

The slope of the curve at point ;

Hence slope of the tangent to the curve at point ;

Therefore;

The angle between the line  and the tangent to the curve at point ;

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