Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2005  OctNov  (P19709/01)  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 xcoordinate of point of intersection of two lines (or line and the curve) at hand, we can find the ycoordinate of the point of intersection of two lines (or line and the curve) by substituting value of xcoordinate 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 xaxis.
First we find the angle of line from xaxis;
In this case equation of a line becomes;
It can be rewritten as;
SlopeIntercept 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 xcoordinates 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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