Find an equation for the surface of revolution generated by revolving the curve $x z = 3$ in the $x z$-plane about the $z$-axis.
A) $y ^ { 2 } + z ^ { 2 } = \frac { 9 } { x ^ { 2 } }$
B) $z ^ { 2 } + x ^ { 2 } = \frac { 9 } { y ^ { 2 } }$
C)$y ^ { 2 } + z ^ { 2 } = \frac { 3 } { x ^ { 2 } }$
D)$y ^ { 2 } + x ^ { 2 } = \frac { 3 } { z }$
E)$y ^ { 2 } + x ^ { 2 } = \frac { 9 } { z ^ { 2 } }$

Question

Quizplus · Accepted Answer

Initially, we have an equation in the form of $x z = 3$. To get the equation for the surface of revolution, we need to express this equation in terms of $x$ and $z$, which can be done by solving for $z$ as follows:

\begin{align*}
x z &= 3 \
z &= \frac{3}{x}
\end{align*}

Now we have an equation for the variable $z$ in terms of the variable $x$, which we can use to express the equation for the surface of revolution. To do this, we need to use the formula for a surface of revolution generated by revolving a curve about an axis, which is given by:

$$y^2 + z^2 = f(x)^2$$

where $f(x)$ is the equation for the curve being revolved. Substituting $z = \frac{3}{x}$ into this formula gives:

$$y^2 + \left(\frac{3}{x}\right)^2 = f(x)^2$$

where $f(x) = x$. Simplifying this equation gives:

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

Multiplying both sides by $x^2$ gives:

$$x^2 y^2 + 9 = x^4$$

Dividing both sides by $x^4$ gives:

$$\frac{y^2}{x^2} + \frac{9}{x^4} = 1$$

Re-arranging this equation gives the final answer:

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

Find an Equation for the Surface of Revolution Generated by Revolving