Algebra : Line and Curve intersects

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.

IMG_20150525_161847Notice 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

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