493. You have seen that an ellipse is in some ways like a circle. The following exercise, which illustrates an alternative definition, shows that an ellipse is in some ways also like a parabola: Let F = (0, 0) be the focus and the line y = −6 be the directrix. Plot several points P that are half as far from the focus as they are from the directrix. The configuration of all such points is an ellipse. Identify the four points where the ellipse crosses the coordinate axes (two on each axis). Use the distance formula to write an equation for the ellipse.
(Continuation) Notice that this ellipse does not have the origin as its center, unlike the previous examples you have seen, so the x-axis is not a line of symmetry. The value of a for this ellipse should be clear, and the value of b can be obtained by finding where the ellipse meets its minor axis. Now calculate the value of c and notice that, had this ellipse been drawn by the string method, one end of the string would be held at the origin F.
To find the four points where the ellipse crosses the coordinate axes, we can consider points on the ellipse where the y-coordinate is 0 and the x-coordinate is 0.
When the y-coordinate is 0, we have y = 0 in the equation of the ellipse. Let's substitute this into the equation of the ellipse to solve for x:
(x - 0)^2 + (0 + 6)^2 = (1/2)(x - 0)^2
x^2 + 36 = (1/2)x^2
2x^2 + 72 = x^2
x^2 = 72
x = ± √(72)
x = ± 6√2
So the ellipse crosses the x-axis at the points (-6√2, 0) and (6√2, 0).
When the x-coordinate is 0, we have x = 0 in the equation of the ellipse. Let's substitute this into the equation of the ellipse to solve for y:
(0 - 0)^2 + (y + 6)^2 = (1/2)(0 - 0)^2
y^2 + 36 = (1/2)y^2
2y^2 + 72 = y^2
y^2 = 72
y = ± √(72)
y = ± 6√2
So the ellipse crosses the y-axis at the points (0, -6√2) and (0, 6√2).
The distance formula for an ellipse is given by:
d = √((x - h)^2/a^2 + (y - k)^2/b^2)
where (h, k) is the center of the ellipse. From the problem description, we know that F = (0, 0) is the focus and the line y = -6 is the directrix. Since the directrix is a horizontal line and the focus is at the origin, the center of the ellipse is (0, 3). We also know that the ellipse crosses its minor axis at (-6√2, 0) and (6√2, 0), so the length of the minor axis is 2a, where a is the distance from the center to one of the points on the minor axis.
Let's find the value of a:
a = (1/2) * (distance from center to point on minor axis)
a = (1/2) * (6√2 - 0)
a = 3√2
Now we can write the equation for the ellipse:
d = √((x - 0)^2/(3√2)^2 + (y - 3)^2/b^2)
To find the value of b, we can use the distance between the focus and the center of the ellipse:
c = √(a^2 - b^2)
0 = c
√(72) = √(18b^2 - b^4)
72 = 18b^2 - b^4
b^4 - 18b^2 + 72 = 0
(b^2 - 12)(b^2 - 6) = 0
This gives us two possible values for b: b = √12 or b = √6.
So the equation for the ellipse is:
d = √(x^2/(18) + (y - 3)^2/(12))