Find-the-equation-of-the-line-that-is-tangent-to-the-curve-y-x-3-and-is-parallel-to-the-line-3x-y-1-0-

Question Number 31812 by NECx last updated on 15/Mar/18

F i n d t h e e q u a t i o n o f t h e l i n e

t h a t i s t a n g e n t t o t h e c u r v e y = x^{3}

a n d i s p a r a l l e l t o t h e l i n e

3 x - y + 1 = 0

Answered by mrW2 last updated on 15/Mar/18

l e t y = 3 x + c b e t h e l i n e t a n g e n t t o y = x^{3}

\frac{d y}{d x} = 3 x^{2} = 3

\Rightarrow x = \pm 1

y = 3 x + c = \pm 3 + c = \pm 1

\Rightarrow c = - 2 o r 2

\Rightarrow e q n . o f l i n e i s

y = 3 x - 2 o r

y = 3 x + 2

Answered by ajfour last updated on 15/Mar/18

l e t e q . o f t a n g e n t t o y = x^{3} a t (x_{1}, x_{1}^{3}) b e

y - x_{1}^{3} = 3 x_{1}^{2} (x - x_{1})

a s \frac{d y}{d x} ∣_{x = x_{1}} = 3 (s l o p e o f g i v e n / / l i n e)

\Rightarrow 3 x_{1}^{2} = 3 \Rightarrow x_{1} = \pm 1

s o r e q d . t a n g e n t e q s . a r e

y - 1 = 3 (x - 1) o r y = 3 x - 2

a n d y + 1 = 3 (x + 1) o r y = 3 x + 2 .

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