Calculus: Analytic Geometry and Calculus, with Vectors

6.3 Ellipses 373

2 The equation

xz^2

22+3z=1

differs from equations of ellipses having their foci on the x axis because the

denominator under xz is not greater than the denominator under yz. Never-

theless, plot the four points on the graph obtained by setting x = 0 and then

y = 0, and then sketch the graph. Observe that everything is like the preceding

problem except that the roles of x and y are interchanged. Then proceed to

find the eccentricity, foci, and directrices. Repeat the process when 2 and 3

are respectively replaced by

(a) 2 and 5 (b) 1 and 5 (c) 3 and 5

3 We have known for a long time that the graph of the equation

(x-h)z+(y-k)z=az

is a circle having its center at the point (h,k). With this hint, sketch graphs of

the equations

(a) (x

52

1)z+(y 22)2=1 (b) (x-1)2 (y-2)z=1

2 22 +^52

Observe that, in these cases, distances from centers to foci are not coordinates
of foci; suitable adjustments must be made. Remark: A good clean start is
made by setting y = 2 and calculating x - 1 and then x.
4 Find the equations of the ellipses (if any) which have foci at the points
(-2,0) and (2,0) and which pass through the point (1,1).
5 Find the equation of the ellipse which has its center at the point (2,3),
which has axes parallel to the coordinate axes, and which is tangent to the coordi-
nate axes. Sketch a reasonably good figure.
6 The foci of a particular ellipse lie midway between the center and vertices.
Find the eccentricity. Supposing that the major axis has length 2a, find the
length of the minor axis and the distance from the center to the directrices.
Sketch a reasonably good figure.
7 Except for minor perturbations, the orbit of the earth is an ellipse having
the sun at a focus. The least and the
greatest distances from the earth to the
sun have the ratio H. Find the ec-
centricity of the approximate orbit.
An r.: Wig.

8 AsinFigure 6. 3 91,letP1(xl,yl)be Pi"''yi 1

a point on the ellipse having the stand-
ard equation

xR^2

az+b =1.

Supposing that Pi is not one of the
points where the ellipse intersects the

Figure 6.391