Resolved: "Carefully describe the level surfaces of h(x, y,..."

High School

Shweta Goel

Answered question

2022-05-22

Carefully describe the level surfaces of h(x, y, z) = x ^2− y^2 − z^2 − 4x + 3.
Your description must include the name and a general equation of the
family of level surfaces.

Answer & Explanation

Jeffrey Jordon

Expert2022-11-08Added 2605 answers

$h (x, y, z) = x^{2} - y^{2} - z^{2} - 4 x + 3$

Let $h (x, y, z) = k$ then

$x^{2} - y^{2} - z^{2} - 4 x + 3 = k$

$x^{2} - 4 x - y^{2} - z^{2} + 3 = k$

$(x^{2} - 4 x + 4 - 4) - y^{2} - z^{2} + 3 = k$

$(x - 2)^{2} - 4 - y^{2} - z^{2} = k - 3$

$(x - 2)^{2} - y^{2} - z^{2} = k + 1$

$\frac{(x - 2)^{2}}{(k + 1)} - \frac{(y^{2})}{(k + 1)} - \frac{(z^{2})}{(k + 1)} = 1$ ..... (1)

Equation (1) is of the type $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1$ (for $k + 1 > 0$ ), know as hyperboloid of two sheets with its axis along $x = 2, y = 0, z = 0$

For $k + 1 < 0$ , $\frac{(x - 2)^{2}}{(k + 1)} - \frac{(y^{2})}{(k + 1)} - \frac{(z^{2})}{(k + 1)} = 1$ $\Rightarrow$ $\frac{y^{2}}{{- (k + 1)}} + \frac{(z^{2})}{{- (k + 1)}} - \frac{{(x - 2)}^{2}}{(k + 1)} = 1$

is hyperboloid of one sheet with its axis along the line $x = 2, y = 0, z = 0$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High School

Didn't find what you were looking for?

Store	Original Price of Aquarium ($)
Pets Plus	118
Pet Planet	110