Question

Consider the following equation of an elliptic paraboloid.

x = 9 - y^2 - 9z^2
a. Find the intercepts with the three coordinate​ axes, if they exist.
b. Find the equations of the​ xy-, xz-, and​ yz-traces, if they exist.
c. Sketch a graph of the surface.

Answer 1

a)The intercepts are x(9,0,0), y(0,+3,0)(0,-3,0) and z-axis(0,0,-1)(0,0,1).

b) Equation of traces are:

For xy trace : x = 9 - y²

For xz- trace: x = 9-9z²

For yz-trace: y²+9z² = 9

How the intercepts and trace equations are determined.

Given

x = 9-y²-9z²

a) To find the intercepts

For x-axis

y = 0, z = 0, substitute into the equation

x = 9 - (0)² - 9(0)²

x = 9

For y-axis

Let x = 0, z = 0 and substitute

0 = 9-y²-9(0)²

y = √9

= +-3

For z-axis

x = 0, y = 0 and substitute

0 = 9 - 0²-9z²

z² = -9/-9

z = √1 = +-1

The intercepts are x(9,0,0), y(0,+-3,0) and z-axis(0,0,+-1)

b) Equation of the traces

For xy-trace

let z = 0

x = 9-y²-9(0)²

x = 9 - y²

For xz- trace

let y = 0

x = 9-0² -9z²

x = 9-9z²

For yz-trace

let x = 0

0 = 9 -y²-9z²

y²+9z² = 9

c) The given equation is

x = 9 -y² - 9z² If the general equation of a elliptical paraboloid is

x = a -by²-cz²

where

a, b and c are constants.

a=9, b = -1 and c = -9

The two dimensional equation is plotted