The equation of common tangent to the curves y=6−x−x2 and xy=x+3 is

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The equation of common tangent to the curves $y = 6 - x - x^{2}$ and $x y = x + 3$ is

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$2 x + y = 4$

$3 x + y = 7$

$3 x + y = 10$

$3 x - y = 8$

answer is D.

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Detailed Solution

Let $y = m x + c$ is tangent to $y = 6 - x - x^{2}$ and $x y = x + 3$ then
$m x + c = 6 - x - x^{2}$ and $x (m x + c) = x + 3$
equations have equal roots ${(m + 1)}^{2} = 4 (c - 6)$ and ${(c - 1)}^{2} = - 12 m$
By solving $m = - 3, c = 7$ $∴ y = - 3 x + 7$