For parametric equations x= a cos t and y= b sin t, describe how the values of a and b determine which conic section will be traced

In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The four types of conic section are the hyperbola, the parabola, the circumference and the ellipse.

For the problem we have this parametric equation:

(1) 

From geometry, we know that we can express a circumference in terms of parameters like this:

(2)

being r the radius of the circumference.

On the other hands, we know that a ellipse can be expressed in terms of parameters like this:

(3)

Therefore, we will have three answers that are the cases for the values a and b, namely.

Case 1: Circumference

To the case of a circumference, the more simple ordinary equation is given by:

(4)

Substituting (1) into (4):



But because of the equation (2), necessarily:



Case 2: Ellipse (focal axis matches the x-axis) 

In this case, the simple ordinary equation is given by:

(5) 

being a and b semi-major axis and semi-minor axis respectively. 

Given that a an b are variables of the parametrization, and a and b are variables of the ellipse as well, to avoid confusion we will modify the equation (5) like this:

(6) 

So, substituting (2) into (6):



Necessarily:

  and   

and given that the focal axis matches the x-axis, then:

 

Case 3: Ellipse (focal axis matches the y-axis) 

In this case, applying the same previous reasoning, the simple ordinary equation is given by:

(7)

being a' and b' semi-major axis and semi-minor axis respectively. 

So, substituting (2) into (7):



Necessarily:

  and   

and given that the focal axis matches the y-axis, then:

 

Finally, the conclusions are:

1. If  then a circumference will be traced. (See Figure 1)

2. If  then a ellipse will be traced with focal axis matching the x-axis. (See Figure 2)

3. If  then a ellipse will be traced with focal axis matching the y-axis. (See Figure 3)

the answer is 21%.