Answered: y P(x, y) -a Τ -a

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter1: Equations, Inequalities, And Mathematical Modeling

Section1.3: Modeling With Linear Equations

Problem 61E

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Question

Let T be the tangent line at the point P(x, y) on the graph of the curve x^2/3 + y^2/3 = a^2/3 , a > 0, as shown in the figure. Show that the radius of curvature at P is three times the distance from the origin to the tangent line T.

Expert Solution

Step 1

Concept:

The function f(x) at the point x=a, is a line that just touches the graph of the function at the point in question and is parallel to the graph at that point is called tangent line. The slope of a curved function at a particular point on the curve is found using tangent line.

Step 2

Given:

Radius of curvature

$x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}, a > 0$

$x$

Step 3

The given curve is

$x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}, a > 0$

Differentiating both the sides with respect to $x$

$\frac{2}{3} x^{\frac{- 1}{3}} + \frac{2}{3} y^{\frac{- 1}{3}} \frac{d y}{d x} = 0 \frac{d y}{d x} = \frac{- y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Step 4

Again differentiating both sides with respect to $x$

$\begin{array}{rcl} \frac{d^{2} y}{d x^{2}} & = & \frac{- x^{\frac{1}{3}} (\frac{1}{3}) y^{\frac{- 2}{3}} \frac{d y}{d x} + (\frac{1}{3}) y^{\frac{1}{3}} x^{\frac{- 2}{3}}}{x^{\frac{2}{3}}} \\ = & \frac{\frac{1}{3} (\frac{x^{\frac{1}{3}}}{y^{\frac{2}{3}}} . \frac{y^{\frac{1}{3}}}{x^{\frac{2}{3}}}) + \frac{1}{3} (\frac{y^{\frac{1}{3}}}{x^{\frac{2}{3}}})}{x^{\frac{2}{3}}} \\ = & \frac{1}{3} \frac{(x^{\frac{2}{3}} + y^{\frac{2}{3}})}{y^{\frac{1}{3}} x^{\frac{2}{3}} x^{\frac{2}{3}}} \\ = & \frac{a^{\frac{2}{3}}}{3 y^{\frac{1}{3}} x^{\frac{4}{3}}} \end{array}$

Step 5

Let the point p be $(x_{0}, y_{0})$ then then curvature at p is,

$k = \frac{|{(\frac{d^{2} y}{d x^{2}})}_{(x_{0}, y_{x})}|}{{[1 + {(\frac{d y}{d x})}^{2}_{(x_{0}, y_{0})}]}^{\frac{3}{2}}} o n s i m p l y f y i n g k b e c o m e s a s f o l l o w s, k = \frac{1}{3 a^{\frac{1}{3}} x^{\frac{1}{3}} y^{\frac{1}{3}}}$

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