Using the optimization methods discussed in this chapter find thex2y?= 1 from the point (5,5).16closest and the farthest point on the ellipse_Please give exact values

Given equation of ellipse is, x29+y216=1 Let the pointx,y lies on the ellipse. So the distance from…

Answered: Using the optimization methods…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Using the optimization methods discussed in this chapter find the
x2
y?
= 1 from the point (5,5).
16
closest and the farthest point on the ellipse
_
Please give exact values

Expert Solution

Step 1

Given equation of ellipse is,

$\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1$

Let the point $(x, y)$ lies on the ellipse.

So the distance from the point $(5, 5)$ to the point $(x, y)$ given by the formula is,

$d = \sqrt{{(x - 5)}^{2} + {(y - 5)}^{2}}$

let us assume the function be $f (x)$ .

If we will optimize $f (x)$ then our distance 'd' will also optimized.

Step 2

Let $g (x) = \frac{x^{2}}{9} + \frac{y^{2}}{16} - 1$ be the constraints.

Now by Lagrange's method of optimization,

$\nabla f = λ \nabla g i . e, f_{x} = λ \nabla g_{x} a n d f_{y} = λ g_{y} . . . . . (1)$

Where $f_{x} a n d g_{x}$ are partial derivatives of f and g with respect to x and similarly $f_{y} a n d g_{y}$ are partial derivatives with respect to y.

Step 3

Find the first order of the obtained equation.

$f_{x} = 2 (x - 5) a n d g_{x} = \frac{2 x}{9} f_{y} = 2 (y - 5) a n d g_{y} = \frac{2 y}{16} T h e n f r o m e q u a t i o n (1) 2 (x - 5) = λ \cdot \frac{2 x}{9} 18 x - 90 = λ \cdot 2 x λ = 9 - \frac{45}{x}$

Also,

$2 (y - 5) = λ \cdot \frac{2 y}{16} 16 y - 80 = λ y λ = 16 - \frac{80}{y}$

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