Find an equation of the tangent line to the curve2(a2 + y)² = 25(x² – y?)(a lemniscate) at the point (3, 1).An equation of the tangent line to the lemniscate at the given point is

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Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter6: Applications Of The Derivative

Section6.3: Implicit Differentiation

Problem 24E

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Question

Find an equation of the tangent line to the curve
2(a2 + y)² = 25(x² – y?)
(a lemniscate) at the point (3, 1).
An equation of the tangent line to the lemniscate at the given point is

Expert Solution

Step-1

$G i v e n t h a t 2 {(x^{2} + y^{2})}^{2} = 25 (x^{2} - y^{2}) T o f i n d : e q u a t i o n o f \tan g e n t l i n e a t p o i n t (3, 1) S o l u t i o n : 2 {(x^{2} + y^{2})}^{2} = 25 (x^{2} - y^{2}) 2 \times 2 (x^{2} + y^{2}) (2 x + 2 y \frac{d y}{d x}) = 25 (2 x - 2 y \frac{d y}{d x}) p u t p o i n t (3, 1) 4 (3^{2} + 1^{2}) (2 \times 3 + 2 \times 1 \frac{d y}{d x}) = 25 (2 \times 3 - 2 \times 1 \frac{d y}{d x}) 40 (6 + 2 \frac{d y}{d x}) = 25 (6 - 2 \frac{d y}{d x}) 240 + 80 \frac{d y}{d x} = 150 - 50 \frac{d y}{d x} 130 \frac{d y}{d x} = - 90 \frac{d y}{d x} = - \frac{9}{13}$