Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.
(2,0), (0,3/2), (-2,0), (0,-3/2)
according to your given:
a = 2, b = 3/2
from that just form the equation into the standard form that you must have
learned.
To find the standard form of the equation of an ellipse, we need to know the coordinates of the center and either the lengths of the major and minor axes or the lengths of the semi-major and semi-minor axes.
Given that the center of the ellipse is at the origin (0,0), we can determine the lengths of the semi-major and semi-minor axes by considering the distances from the center to the three given points on the ellipse: (2,0), (0,3/2), and (-2,0).
The distance formula is used to find the distance between two points in a coordinate plane. In this case, the formula is sqrt((x2 - x1)^2 + (y2 - y1)^2).
1) Distance from the center to (2,0):
d = sqrt((2 - 0)^2 + (0 - 0)^2)
= sqrt(4 + 0)
= sqrt(4)
= 2
2) Distance from the center to (0,3/2):
d = sqrt((0 - 0)^2 + (3/2 - 0)^2)
= sqrt(0 + 9/4)
= sqrt(9/4)
= 3/2
3) Distance from the center to (-2,0):
d = sqrt((-2 - 0)^2 + (0 - 0)^2)
= sqrt(4 + 0)
= sqrt(4)
= 2
We have found that the semi-major axis of the ellipse is 2 and the semi-minor axis is 3/2.
Now, we can write the standard form of the equation of the ellipse, assuming a horizontal major axis. The equation is:
x^2/a^2 + y^2/b^2 = 1
where a represents the semi-major axis and b represents the semi-minor axis.
Substituting the values, we get:
x^2/2^2 + y^2/(3/2)^2 = 1
x^2/4 + y^2/(9/4) = 1
x^2/4 + (4/9)y^2 = 1
Therefore, the standard form of the equation of the given ellipse with the center at the origin is:
x^2/4 + (4/9)y^2 = 1