Question

Find the equation of the common tangent to the curves y = 6 – x – x2 and xy = x + 3.

Answer 1

The given curves are y = 6 – x – x2 and y = 1 + 3/x

⇒ dy/dx = –1 – 2x and dy/dx = – 3/x2

Since the given curves have their common tangent,

so – 3/x2 = –1 – 2x

⇒ 2x3 + x2 – 3 = 0

⇒ x = 1

when x = 1, y = 6 – 1 – 1 = 4

So, the point is (1, 4)

Hence, the equation of the common tangent is

y – 4 = – 3 (x – 1)

⇒ y – 4 = – 3x + 3

⇒ 3x + y = 7.