Find a parametric description for the ellipse having the focus F=(0,-4), corresponding directrix y=3, and eccentricity e=3/4.
To find a parametric description for the given ellipse, we need to find the equation of the ellipse and then express it in parametric form.
Let's start by finding the equation of the ellipse:
1. Recall that the eccentricity (e) of an ellipse is defined as the ratio of the distance between the focus (F) and any point (P) on the ellipse to the distance between that point (P) and the directrix.
For our ellipse, we have:
Focus (F) = (0, -4)
Directrix: y = 3
Eccentricity (e) = 3/4
2. The distance between the focus (F) and any point on the ellipse (P) can be expressed as: |PF| = e * |PD|, where |PD| is the perpendicular distance between the point (P) and the directrix.
In our case, since the directrix is a horizontal line in the form y = constant, the perpendicular distance |PD| is given by the absolute difference between the y-coordinates of point (P) and the directrix:
|PD| = |y - 3|
3. Substituting the given values, we have the equation: |PF| = (3/4) * |y - 3|
4. Squaring both sides to eliminate the absolute value, we get:
(x - 0)^2 + (y + 4)^2 = [(3/4) * (y - 3)]^2
Simplifying this equation, we have:
x^2 + (y + 4)^2 = (9/16) * (y - 3)^2
5. Expanding and rearranging the terms, we obtain:
x^2 + y^2 + 8y + 16 = (9/16) * (y^2 - 6y + 9)
Multiplying through by 16 to clear the fraction, we get:
16x^2 + 16y^2 + 128y + 256 = 9y^2 - 54y + 81
6. Combining like terms, we have:
16x^2 + 7y^2 + 182y + 175 = 0
This is the equation of the ellipse.
Now, let's express this equation in parametric form:
Let x = a * cos(t) and y = b * sin(t), where a and b are positive constants.
Substituting these values into the equation of the ellipse, we have:
16(a * cos(t))^2 + 7(b * sin(t))^2 + 182(b * sin(t)) + 175 = 0
Simplifying this equation, we obtain:
16a^2 * cos^2(t) + 7b^2 * sin^2(t) + 182b * sin(t) + 175 = 0
Now, we can choose suitable values for a and b to make this equation satisfy the given ellipse. However, there are multiple possible solutions for a and b, which would result in different parametric descriptions for the ellipse.
The most common set of values for a and b is a = 4 and b = 5. Substituting these values into the equation, we have:
64 * cos^2(t) + 175 * sin^2(t) + 910 * sin(t) + 175 = 0
Therefore, the parametric description for the ellipse is:
x = 4 * cos(t)
y = 5 * sin(t)
Note: Different choices of a and b will yield different parametric descriptions of the same ellipse.