Math - Matching Quadric Equations Surfaces Quiz

1. What is the equation of an ellipsoid in 3D space?

X^2/a^2 - y^2/b^2 + z^2/c^2 = 1

X^2/a + y^2/b + z^2/c = 1

X^2/a^2 + y^2/b^2 + z^2/c^2 = 1All Traces are ellipsesIf a=b=c, sphere

X/a + y/b + z/c = 1

An ellipsoid in 3D space is defined by the equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1. This equation signifies that all traces of an ellipsoid are ellipses. If a=b=c, then the ellipsoid becomes a sphere.

Explanation

An ellipsoid in 3D space is defined by the equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1. This equation signifies that all traces of an ellipsoid are ellipses. If a=b=c, then the ellipsoid becomes a sphere.

2. Cone

Z^2/c^2 = x^2/a^2 + y^2/b^2Horizontal Traces are ellipsesVertical traces in planes x=k and y=k are hyperbolas if k is not 0, a pair of lines if k=0

Z^2/c = x/a + y/b. This equation results in a plane when graphed.

Z^2/c^2 = x^2/a^2 - y^2/b^2. The traces result in parabolas in this case.

Z/c = x^2/a - y^2/b. The traces will form circles in this scenario.

The correct equation for a cone is z^2/c^2 = x^2/a^2 + y^2/b^2. This equation results in different types of traces based on the values of a, b, and c as mentioned. Each of the incorrect answers provided lead to different shapes when graphed, demonstrating the importance of using the correct equation for the shape being studied.

Explanation

The correct equation for a cone is z^2/c^2 = x^2/a^2 + y^2/b^2. This equation results in different types of traces based on the values of a, b, and c as mentioned. Each of the incorrect answers provided lead to different shapes when graphed, demonstrating the importance of using the correct equation for the shape being studied.

3. What is the equation for an Elliptic Paraboloid?

Z/c= x^2/a^2 + y^2/b^2Horizontal Traces are ellipsesvertical traces are parabolasVariable raised to the first power indicates axis of paraboloid

Z = x^2 - y^2

Z = x^2/a^2 - y^2/b^2

Z = x^2 + y^2

An elliptic paraboloid is defined by the equation z/c = x^2/a^2 + y^2/b^2, where horizontal traces are ellipses, vertical traces are parabolas, and a variable raised to the first power indicates the axis of the paraboloid.

Explanation

An elliptic paraboloid is defined by the equation z/c = x^2/a^2 + y^2/b^2, where horizontal traces are ellipses, vertical traces are parabolas, and a variable raised to the first power indicates the axis of the paraboloid.

4. What is the equation of a Hyperboloid of one sheet?

X^2/a^2 - y^2/b^2 - z^2/c^2 = 1

X^2/a^2 - y^2/b^2 + z^2/c^2 = 1

X^2/a^2 + y^2/b^2 - z^2/c^2 = 1Horizontal traces are ellipses vertical traces are hyperbolasaxis of symmetry corresponds with variable whose coefficient is negative

X^2/a^2 + y^2/b^2 + z^2/c^2 = 1

The correct equation for a Hyperboloid of one sheet is x^2/a^2 + y^2/b^2 - z^2/c^2 = 1. This equation represents a hyperboloid where horizontal traces are ellipses, vertical traces are hyperbolas, and the axis of symmetry corresponds with the variable whose coefficient is negative.

Explanation

The correct equation for a Hyperboloid of one sheet is x^2/a^2 + y^2/b^2 - z^2/c^2 = 1. This equation represents a hyperboloid where horizontal traces are ellipses, vertical traces are hyperbolas, and the axis of symmetry corresponds with the variable whose coefficient is negative.

5. What is the equation for a Hyperbolic Paraboloid?

Z/c= x^2/a^2 - y^2/b^2Horizontal traces are hyperbolasVertical traces are parabolas

Z = x^2/a^2 + y^2/b^2Horizontal traces are parabolasVertical traces are hyperbolas

Z = x^2/a^2 - y^2/b^2Horizontal traces are circlesVertical traces are ellipses

Z = x^2/a^2 + y^2/b^2Horizontal traces are ellipsesVertical traces are circles

A hyperbolic paraboloid is a doubly-ruled surface shaped like a saddle. The correct equation for it involves the difference of squares, leading to hyperbolic parabolas in horizontal traces and parabolas in vertical traces.

Explanation

A hyperbolic paraboloid is a doubly-ruled surface shaped like a saddle. The correct equation for it involves the difference of squares, leading to hyperbolic parabolas in horizontal traces and parabolas in vertical traces.

6. What geometric shape is represented by the equation -x^2/a^2 - y^2/b^2 + z^2/c^2 = 1?

-x^2/a^2 - y^2/b^2 + z^2/c^2 = 1Horizontal traces in z=k are ellipses if k>c or k<-cVertical Traces are hyperbolasTwo minus signs indicate two sheets

Cone.

Ellipsoid.

Cylinder.

The correct answer represents the Hyperboloid of two sheets, which is a three-dimensional surface defined by the given equation. The other options are different geometric shapes and do not match the provided equation.

Explanation

The correct answer represents the Hyperboloid of two sheets, which is a three-dimensional surface defined by the given equation. The other options are different geometric shapes and do not match the provided equation.