Question

Bridgette has drawn a representation of a sphere with a

radius of 3 units, using the graphs of a circle and an ellipse,
as shown in the picture below on the left.
She needs to determine the equations of the graphs that
will generate the representation of the 3D sphere, as
shown in the picture below on the right.
What are possible equations of the circle and ellipse that
Bridgette can use to create her 3D drawing?

Answer 1

C

Explanation:

You want the equations of the circle and ellipse in the given figure.

Equations

The standard form equation for an ellipse is ...

`(x^2)/(a^2)+(y^2)/(b^2)=1`

where 'a' and 'b' are the semi-axes in the x-direction and y-direction, respectively.

You will notice that when a=b=r, the ellipse becomes a circle and the equation can be rewritten as ...

`(x^2)/(r^2)+(y^2)/(r^2)=1\ \Longleftrightarrow\ x^2+y^2=r^2`

Application

Here, the radius is 3, the same as the semi-major axis in the x-direction (a=3). The ellipse axis in the y-direction appears to be about 1/3 that value, so b=1, and the equations of the circle and ellipse are ...

`x^2+y^2=9\qquad\text{circle}\\\\(x^2)/(9)+(y^2)/(1)=1\qquad\text{ellipse}`