find an equation in standard form for the ellipse if the major axis endpoints are (-5,2) and (3,2); the minor axis length is 6.
major axis is parallel to the y - axis
centre or midpoint is (-1,2) , were
b= 4 and a = 3
(x+1)^2 /9 + (y-2)/16 = 1
To find the equation of an ellipse in standard form, we need the coordinates of the center and the lengths of the major and minor axes.
Given:
Major axis endpoints: (-5, 2) and (3, 2)
Minor axis length: 6
Step 1: Find the center of the ellipse.
The center of the ellipse is the midpoint of the major axis. To find it, we average the x-coordinates and the y-coordinates of the major axis endpoints.
x-coordinate of the center = (x1 + x2) / 2 = (-5 + 3) / 2 = -2/2 = -1
y-coordinate of the center = (y1 + y2) / 2 = (2 + 2) / 2 = 4/2 = 2
So, the center of the ellipse is (-1, 2).
Step 2: Find the lengths of the major and minor axes.
The length of the major axis is the distance between the major axis endpoints.
Length of major axis = distance between (-5, 2) and (3, 2) = 3 - (-5) = 8
The length of the minor axis is given as 6.
Step 3: Write the equation in standard form.
The standard form of the equation of an ellipse is:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
Where:
- (h, k) is the center of the ellipse
- a is the distance from the center to the endpoints of the major axis (half of the major axis length)
- b is the distance from the center to the endpoints of the minor axis (half of the minor axis length)
Plugging in the values we found:
Center: (h, k) = (-1, 2)
Major axis length: 2a = 8 (so, a = 4)
Minor axis length: 2b = 6 (so, b = 3)
Now, we can write the equation:
((x + 1)^2 / 4^2) + ((y - 2)^2 / 3^2) = 1
Simplifying:
(x + 1)^2 / 16 + (y - 2)^2 / 9 = 1
Therefore, the equation of the ellipse in standard form is ((x + 1)^2 / 16) + ((y - 2)^2 / 9) = 1.