Hyperboloid with different sheets?
I just took a quiz and have a couple basic questions because I am too anxious to wait a week for my score.
My questions are:
1) *I put hyperboloid of one sheet for this $$ \frac{(x-1)^{2}}{2^{2} } - \frac{(y)^{2}}{2^{2}} + \frac{(z)^{2}}{2^{2}} = 1 $$
2) *I put hyperboloid of one sheet again, but feeling like it's two sheets. $$ \frac{-(x)^{2}}{2^{2} } + \frac{(y)^{2}}{2^{2}} + \frac{(z)^{2}}{2^{2}} + 10 = 0 $$
$\endgroup$ 11 Answer
$\begingroup$The general equation for a hyperboloid of one sheet is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$$ For $1)$, the equation is precisely this form (with the roles of $y$ and $z$ reversed so that the axis of symmetry is the $y$ axis). The general equation for a hyperboloid of two sheets is $$\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$$ Your equation is $$\frac{-x^2}{2^2} + \frac{y^2}{2^2} + \frac{z^2}{2^2} + 10= 0 \implies \frac{x^2}{10*2^2} - \frac{y^2}{10*2^2} - \frac{z^2}{10*2^2} = 1$$ shiwch is indeed an equation of a hyperboloid of two sheets (comparing it with the general equation).
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