Write an equation in standard form of the ellipse described. The center of the ellipse is the origin.
Focus at (0, 6); vertex at (0, 7)
the major axis is vertical, and we have
a = 7
c = 6
b^2 = a^2-c^2 = 13
x^2/13 + y^2/49 = 1
To determine the equation of the ellipse in standard form, we need to find the values of a and b.
For an ellipse with the center at the origin, the equation in standard form is:
x^2 / a^2 + y^2 / b^2 = 1
The distance from the center to each focus point is c, while the distance from the center to each vertex is a.
Given that the focus is at (0, 6) and the vertex is at (0, 7), we can determine the value of a:
a = 7 - 0 = 7
Now, let's find the value of c:
c = 6 - 0 = 6
With the values of a and c, we can use the following relationship to find b:
c^2 = a^2 - b^2
b^2 = a^2 - c^2
b^2 = 7^2 - 6^2
b^2 = 49 - 36
b^2 = 13
Now we have the values of a and b. Substituting these values into the equation, we get:
x^2 / 7^2 + y^2 / 13 = 1
Simplifying, we have:
x^2 / 49 + y^2 / 13 = 1
Therefore, the equation of the ellipse in standard form is:
x^2 / 49 + y^2 / 13 = 1
To write the equation of an ellipse in standard form, we need to know the coordinates of the center and the lengths of the major and minor axes, or the distances from the center to the foci and vertices.
In this case, we are given that the center of the ellipse is at the origin (0,0). We are also given that the focus is at (0,6) and the vertex at (0,7).
Since the focus and the vertex both lie on the major axis of the ellipse, the distance between the center and the focus is the same as the distance between the center and the vertex. This distance is called the semi-major axis. In this case, the semi-major axis is 7.
The equation of an ellipse with the center at the origin is:
x^2/a^2 + y^2/b^2 = 1
where a is the length of the semi-major axis and b is the length of the semi-minor axis.
In this case, the semi-major axis a is 7.
To find the length of the semi-minor axis b, we can use the relationship between the semi-major axis a, the semi-minor axis b, and the distance between the center and the focus c, which is known as the eccentricity of the ellipse.
The eccentricity (e) is defined as the ratio of the distance between the center and the focus (c) to the length of the semi-major axis (a). For an ellipse, the eccentricity is always less than 1.
Using the given coordinates of the focus (0,6) and the vertex (0,7), we can calculate the distance between the center and the focus:
c = 6
Since the center of the ellipse is at the origin, the distances between the vertices and the center along the major axis are equal. Therefore, the distance between the center and the vertex is also 7.
Using the formula for eccentricity, c/a = e, we can solve for e:
6/7 = e
To find b, the length of the semi-minor axis, we can use the relationship between the semi-major axis a, the semi-minor axis b, and the eccentricity e:
e^2 = 1 - (b^2/a^2)
Plugging in the values we know:
(6/7)^2 = 1 - (b^2/7^2)
Solving for b:
(36/49) = 1 - (b^2/49)
36/49 = 49/49 - (b^2/49)
b^2/49 = 13/49
b^2 = 13
Taking the square root of both sides:
b = √(13)
Now we can write the equation of the ellipse in standard form:
x^2/7^2 + y^2/(√13)^2 = 1
Simplifying:
x^2/49 + y^2/13 = 1
Therefore, the equation in standard form of the given ellipse with the center at the origin, the focus at (0,6), and the vertex at (0,7) is:
x^2/49 + y^2/13 = 1