Application of Derivatives

Find the equation of the tangent to the curve x 2 + 3y = 3, which is parallel to the line y – 4x + 5 = 0. - Wired Faculty

Question

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Find the equation of the tangent to the curve x² + 3y = 3, which is parallel to the line y – 4x + 5 = 0.

Solution

The equation of curve is x² + 3y = 3 ...(1)
Differentiating w.r.t.x, we get,
$2 straight x plus 3 dy over dx space equals 0 space space space space rightwards double arrow space space space space dy over dx space equals space minus fraction numerator 2 straight x over denominator 3 end fraction therefore space space space straight m subscript 1 space equals space minus fraction numerator 2 straight x over denominator 3 end fraction Let space space straight m subscript 2 space be space slope space of space line space space straight y minus 4 straight x plus 5 space equals space 0 therefore space space space space space space straight m subscript 2 space equals 4 because space space space tangent space is space parallel space to space straight y minus 4 straight x plus 5 space equals space 0 therefore space space space space space space straight m subscript 1 space equals space straight m subscript 2 therefore space space space space space space space fraction numerator negative 2 straight x over denominator 3 end fraction space equals space 4 space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight x space equals space minus 6 therefore space space space space space space from space left parenthesis 1 right parenthesis comma space space space 36 plus 3 straight y space equals space 3 space space space space rightwards double arrow space space space space straight y space equals space minus 11 therefore space space space point space is space left parenthesis negative 6 comma space minus 11 right parenthesis therefore space space space space equation space of space tangent space at space left parenthesis negative 6 comma space minus 11 right parenthesis space is space space space space space straight y plus 11 space equals space 4 left parenthesis straight x plus 6 right parenthesis space space space space or space space space space space space straight y plus 11 space equals space 4 straight x plus 24 space space space space space or space space space space 4 straight x minus straight y plus 13 space equals space 0$

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