Find all points on the graph of x2y − 2x + 63y = 2 where the tangent line is horizontal. (Enter your answer as a comma-separated list of ordered pairs, (x, y). If an answer does not exist, enter DNE.)

Find all points on the graph of x2y − 2x + 63y = 2 where the tangent line is horizontal. (Enter your answer as a comma-separated list of ordered pairs, (x, y). If an answer does not exist, enter DNE.)

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.2: Introduction To Conics: parabolas

Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.

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Question

Find all points on the graph of

x²y − 2x + 63y = 2

where the tangent line is horizontal. (Enter your answer as a comma-separated list of ordered pairs, (x, y).

If an answer does not exist, enter DNE.)

Expert Solution

Step 1

Differentiate the equation $x^{2} y - 2 x + 63 y = 2$ wrt x:

$\frac{d}{d x} x^{2} y - \frac{d}{d x} 2 x + \frac{d}{d x} 63 y = \frac{d}{d x} 2 2 x y + x^{2} \frac{d y}{d x} - 2 + 63 \frac{d y}{d x} = 0$

Step 2

Since tangent is horizontal so $\frac{d y}{d x} = 0$ . Thus, the equation $2 x y + x^{2} \frac{d y}{d x} - 2 + 63 \frac{d y}{d x} = 0$ becomes: $2 x y - 2 = 0$ or $x y = 1$ .

Substitute $y = \frac{1}{x}$ in $x^{2} y - 2 x + 63 y = 2$ :

$x^{2} . \frac{1}{x} - 2 x + \frac{63}{x} = 2 x^{2} - 2 x^{2} + 63 = 2 x x^{2} + 2 x - 63 = 0 (x - 7) (x + 9) = 0 x = 7, - 9$

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