Find the equation of tangent and normal to the curve x = 2 a t 2 1 + t 2 , y = 2 a t 3 1 + t 2 at the point for which t = 1 2 .

Given: x = 2 a t 2 1 + t 2 , y = 2 a t 3 1 + t 2 At t = 1 2 , x = 2 a 5 , y = a 5 Also, d x d t = 4 a t 1 + t 2 2 and d y d t = 2 a t 2 3 + t 2 1 + t 2 2 ∴ d y d x = d y d t d x d t = 1 2 t 3 + t 2 Slope of tangent at t = 1 2 is d y d x t = 1 2 = 1 2 1 2 3 + 1 4 = 13 16 Slope of normal at t = 1 2 is - d x d y t = 1 2 = - 16 13 . Equation of tangent when t = 1 2 is y - a 5 = 13 16 x - 2 a 5 ⇒ 13 x - 16 y = 2 a And, the equation of normal is y - a 5 13 16 + x - 2 a 5 = 0 ⇒ 16 x + 13 y = 9 a

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 5

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Earn 100

Number of distinct real root(s) of the equation $- x^{3} + 6 x^{2} - 18 x + 4 = 0$ is/are

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Important Questions on Application of Derivatives

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 6

Number of distinct real values of

x

satisfying the equation

x^{3} - 5 x^{2} + 7 x - 3 = 0

is/are

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 7

The equation

x^{4} - 2 x^{2} = a

has

4

distinct real roots, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 8

Statement - 1: If $y = f (x)$ is increasing in $[α, β],$ then its range is $[f (α), f (β)]$

Statement - 2: Every increasing function need not be continuous.

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 9

Draw the graph of $y = - 3 x^{4} + 4 x^{3} + 12 x^{2} - 2$ .

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 10

The graph of

y = x^{4} - 2 x^{2} + 5

is

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 11

Find the equation of tangent to curve $y = 3 x^{2} + 4 x + 5$ at $(0, 5)$

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 12

Find the equation of tangent and normal to the curve $x^{2} + 3 x y + y^{2} = 5$ at point $(1, 1)$ on it.

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 13

Find the equation of tangent and normal to the curve $x = \frac{2 a t^{2}}{1 + t^{2}}, y = \frac{2 a t^{3}}{1 + t^{2}}$ at the point for which $t = \frac{1}{2}$ .