Find the equation of the ellipse having the center at the origin and the major axes on OX.
a) passing through (2 sq root of 2, -2), distance between foci is (4 sq root of 2)
b)passing through ( -2,4), vertex at (4, 0)
To find the equation of an ellipse given the center and information about its major axes, we can follow these steps:
Step 1: Determine the center of the ellipse.
Since the center of the ellipse is given to be at the origin (0, 0), the center is (h, k) = (0, 0).
Step 2: Find the value of 'a' (semi-major axis).
In an ellipse, the semi-major axis is the distance from the center to the farthest point on the ellipse along the major axis.
a) For case (a), the distance between the center (0, 0) and the point (2√2, -2) is equal to 'a'.
Therefore, a = √((2√2)^2 + (-2)^2) = √(8 + 4) = √12 = 2√3.
b) For case (b), the distance between the center (0, 0) and the vertex (4, 0) is equal to 'a'.
Therefore, a = 4.
Step 3: Find the value of 'b' (semi-minor axis).
In an ellipse, the semi-minor axis is the distance from the center to the farthest point on the ellipse along the minor axis.
To find 'b', we can use the formula c^2 = a^2 - b^2, where c represents the distance between the foci. Rearranging the formula, we can solve for 'b' as b = √(a^2 - c^2).
a) For case (a), the distance between the foci is given as 4√2.
c = 4√2 / 2 = 2√2.
b = √((2√3)^2 - (2√2)^2) = √(12 - 8) = √4 = 2.
b) For case (b), the distance between the foci is not provided, so we cannot determine the value of 'b' without additional information.
Step 4: Write the equation of the ellipse.
The equation of an ellipse centered at the origin can be written in the standard form as (x^2 / a^2) + (y^2 / b^2) = 1.
a) For case (a), the equation is (x^2 / (2√3)^2) + (y^2 / 2^2) = 1.
Simplifying, the equation becomes x^2 / 12 + y^2 / 4 = 1.
b) For case (b), without the information about the distance between the foci, we can't determine the equation of the ellipse.