Mathematics, 26.11.2019 16:31 minstcordell4115

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

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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) $\left\{{{x=acost}\atop{y=bsint}}\right$

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

(2) $\left \{ {{x=rcost} \atop {y=rsint}}\right$

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) $\left \{ {{x=acost} \atop {y=bsint}}\right$

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) $x^{2} + y^{2} = r^{2}$

Substituting (1) into (4):

$a^{2}cos^{2}t+b^{2}cos^{2}t=r^{2}$

But because of the equation (2), necessarily:

a = b = r

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

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

(5) $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

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) $\frac{x^{2}}{a'^{2}}+\frac{y^{2}}{b'^{2}}=1$

So, substituting (2) into (6):

$\frac{a^{2}cos^{2}t}{a'^{2}}+\frac{b^{2}cos^{2}t}{b'^{2}}=1$

Necessarily:

a=a' and b=b'

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) $\frac{x^{2} }{b'^{2}}+\frac{y^{2}}{a'^{2}}=1$

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

So, substituting (2) into (7):

$\frac{a^{2}cos^{2}t}{b'^{2}}+\frac{b^{2}cos^{2}t}{a'^{2}}=1$

Necessarily:

a = b' and b = a'

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

Finally, the conclusions are:

1. If a = b 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)

For parametric equations x= a cos t and y= b sin t, describe how the values of a and b determine whi

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