Write an equation of an ellipse centered at the origin, satisfying the given conditions. vertex (0,sqrt (29)); covertex (-5,0)
The standard form equation of an ellipse centered at the origin with the major axis parallel to the y-axis is:
(x^2) / (b^2) + (y^2) / (a^2) = 1
where a is the length of the semi-major axis and b is the length of the semi-minor axis.
The vertex (0,sqrt(29)) is on the major axis, so it corresponds to the point (0,a) in the equation. Therefore,
a = sqrt(29)
The covertex (-5,0) is on the minor axis, so it corresponds to the point (b,0) in the equation. Therefore,
b = 5
Substituting these values into the standard form equation, we get:
(x^2) / (5^2) + (y^2) / (sqrt(29)^2) = 1
Simplifying:
x^2 / 25 + y^2 / 29 = 1
This is the equation of the ellipse centered at the origin, with semi-major axis sqrt(29) and semi-minor axis 5.
To write an equation of an ellipse centered at the origin with the given vertex and covertex, we can start by noting that the major axis of the ellipse is vertical.
The vertex (0, √29) indicates that the major axis is vertical and goes upwards from the center (0, 0). So, the length of the major axis is 2a = 2√29, where a is the semi-major axis.
The covertex (-5, 0) indicates that the minor axis is horizontal and goes to the left. So, the length of the minor axis is 2b = 2(5) = 10, where b is the semi-minor axis.
Therefore, we have:
a = √29
b = 5
The equation of an ellipse centered at the origin is:
(x^2 / a^2) + (y^2 / b^2) = 1
Substituting the values:
(x^2 / (√29)^2) + (y^2 / 5^2) = 1
Simplifying:
(x^2 / 29) + (y^2 / 25) = 1
Thus, the equation of the ellipse centered at the origin with the given conditions is (x^2 / 29) + (y^2 / 25) = 1.