EXERCISE 9: Let the conic such that e=0.5 and the directrix is d : y=4 .
The focus F is
at the origin.
a/ What conics is it?
b/ Where is the focal axis? Sketch approximately to have an idea of the problem. On this
graph you will write your details of other questions.
c/ Find the Cartesian coordinates of V 1 the major vertex between the focus F and the
directrix d .
d/ Find the Cartesian coordinates of V 2 the 2 nd major vertex.
e/ Find the Cartesian coordinates of the Center C of the conics. Then find a , c , k and b .
f/ Give the conic Cartesian standard form equation.
g/ Find its Polar equation.
a/ The conic with e = 0.5 and the directrix y = 4 is an ellipse.
b/ The focal axis is the line passing through the focus F (origin) and
perpendicular to the directrix. Since the focus F is at the origin and the
directrix is y = 4, the focal axis is the y-axis.
Here's an approximate sketch of the conic with the focal axis:
c/ To find the coordinates of the major vertex V1, we need to determine the
point on the ellipse that is closest to the directrix.
Since the directrix is a horizontal line, the major vertex will have coordinates (0, a),
where a represents the semi-major axis.
Given that the directrix is y = 4, the distance between the focus F and the directrix is
equal to the value of a.
Therefore, V1 has the Cartesian coordinates (0, 0.5).
d/ The 2nd major vertex V2 will have the same y-coordinate as V1 but with a
negative x-coordinate.
Hence, V2 has the Cartesian coordinates (-2a, 0.5). Since a = 0.5, the coordinates of V2
are (-1, 0.5).
e/ The center C of the ellipse is the midpoint between the two major vertices
V1 and V2. Therefore, the x-coordinate of C is the average of the x-
coordinates of V1 and V2, and the y-coordinate of C is the average of the y-
coordinates of V1 and V2.
The coordinates of the center C are ((0 + (-1))/2, (0.5 + 0.5)/2) = (-0.5, 0.5).
The value of a is the distance from the center C to either V1 or V2, so a = 0.5.
The value of c is the distance from the center C to the focus F, so c = 0.
The value of k is the distance from the center C to the directrix d, so k = 4.
The value of b can be found using the relationship c^2 = a^2 - b^2. Since c = 0, we have
0 = 0.5^2 - b^2, which gives b = 0.5.
Therefore, the coordinates of the center C are (-0.5, 0.5), a = 0.5, c = 0, k = 4, and b =
0.5.
f/ The conic's Cartesian standard form equation for an ellipse with center
(h, k) and semi-major axis a is:
((x - h)^2)/(a^2) + ((y - k)^2)/(b^2) = 1
In this case, the equation becomes:
((x - (-0.5))^2)/(0.5^2) + ((y - 0.5)^2)/(0.5^2) = 1
Simplifying, we get:
(x + 0.5)^2 + 4(y - 0.5)^2 = 4
g/ The polar equation of an ellipse with center at the origin is given by:
r = (l/(1 - e*cos(theta)))
In this case, since the focus F is at the origin (0, 0) and e = 0.5, the equation becomes:
r = (2/(1 - 0.5*cos(theta)))
