# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#1

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Question

A line has equation and a curve has equation , where c is a constant. Find the set of possible values of c for which the line does not intersect the curve.

Solution

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;      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.

It is evident that if given line and curve do not intersect then solution of above obtained quadratic  equation will not have any real roots. Hence;         