3,Consider Hhe fallauing twe sucfaces2=2xy8x-5y -82= -13a.) Show Hhat the surfaces intersect at the paint Cl, L,2)b.) Shou that the Surfaces have prependicolor tangendplanes at this poln!

Answered: 3. Consider Hhe following two surfaces.…

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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Question

3,
Consider Hhe fallauing twe sucfaces
2=
2xy
8x-5y -82= -13
a.) Show Hhat the surfaces intersect at the paint Cl, L,2)
b.) Shou that the Surfaces have prependicolor tangend
planes at this poln!

Expert Solution

Step 1

Given surfaces are $z = 2 x y^{2}$ and $8 x^{2} - 5 y^{2} - 8 z = - 13$

(a)

Let $f (x, y, z) = z - 2 x y^{2} = 0$

Let $g (x, y, z) = 8 x^{2} - 5 y^{2} - 8 z + 13 = 0$

It is required to show that these two surfaces intersect at $(1, 1, 2)$ .

At $(1, 1, 2)$ , the value of $f (x, y, z) = z - 2 x y^{2}$ is

$f (1, 1, 2) = 2 - 2 (1) {(1)}^{2} = 2 - 2 = 0$

At $(1, 1, 2)$ , the value of $g (x, y, z) = 8 x^{2} - 5 y^{2} - 8 z + 13 = 0$ is

$g (1, 1, 2) = 8 {(1)}^{2} - 5 {(1)}^{2} - 8 (2) + 13 = 0$

Thus $f (1, 1, 2) = g (1, 1, 2) = 0$

Therefore the two surfaces intersect at $(1, 1, 2)$ .

Hence proved that the surfaces intersect at the point $(1, 1, 2)$ .

Step 2

(b)

It is required to show that the surfaces $f (x, y, z) and g (x, y, z)$ have perpendicular tangent planes at $(1, 1, 2)$ .

If the tangent planes to the surfaces are perpendicular, then the gradients(normals) of the surfaces are also perpendicular.

The gradient is given by $\nabla f = \frac{\partial f}{\partial x} \vec{i} + \frac{\partial f}{\partial y} \vec{j} + \frac{\partial f}{\partial z} \vec{k}$

$f (x, y, z) = z - 2 x y^{2} = 0$

$\nabla f = (- 2 y^{2}) \vec{i} + (- 4 x y) \vec{j} + (1) \vec{k}$

At $(1, 1, 2)$ , $\nabla f = - 2 \vec{i} - 4 \vec{j} + 1 \vec{k}$

Therefore ${\nabla f}_{(1, 1, 2)} = - 2 \vec{i} - 4 \vec{j} + 1 \vec{k}$ .......................................(1)

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