Given a line y = x + c and curve y = x² − 4, will they intersect ? Line y = x + c consists of an infinite number of parallel lines depending on value of c. Below shows 3 lines with 3 different c values.
Notice that when c = − 17/4, the line touches the curve at 1 real point. The line is a tangent to the curve. When c = − 8, the line does not intersect the curve. And when c = 0, the line cuts the curve at 2 real points. So to find the points of intersection of a line and curve, we solve the two equations simultaneously. Then the nature of the roots of the quadratic equation depends on the value of discriminant, b² − 4ac. If the value is > 0, there are 2 roots, If the value is = 0, there is only one root and the line is tangent to the curve, and if the value < 0, there are no roots and the line does not intersect the curve.
In the above example, x² − 4 = x + c ⇒ x² − x + (−4 −c) = 0
b² − 4ac ⇒ (−1)² − 4(1)(− 4 −c) = 0 ⇒1 + 16 + 4c = 0
4c = − 17 ∴ c = − 17/4
Try this question :
1) Find the values of c for which y = x + c is tangent to the curve y = 6x − cx² .
Answer:
Solve the two equations simultaneously :
x + c = 6x − cx² ⇒ cx² − 5x + c = 0
For the line to be tangent to the curve ⇒ b² − 4ac = 0 ⇒ (−5)² − 4 (c)(c) = 0 ⇒ 25 − 4c² = 0
∴ c² = 25/4 ⇒ c = ± 5/2
GCE 'O' Level Mathematics 