5) What is the equation in standard form of an ellipse centered at the origin and vertex (0, 4) and focus (0, 1)?
I don’t know how to start the problem off, if you can help with that that would be really helpful
To find the equation of an ellipse in standard form, we need to consider some key properties of the ellipse:
1. The center of the ellipse is at the origin (0, 0).
2. The vertex is (0, 4), which means the distance from the origin to the vertex in the y-direction is 4.
3. The focus is (0, 1), which means the distance from the origin to the focus in the y-direction is 1.
From these properties, we can deduce the following information:
1. The distance from the center to either of the foci is denoted by 'c'.
2. The distance from the center to a vertex is denoted by 'a'.
3. The lengths of the major and minor axes are denoted by '2a' and '2b', respectively.
Using these notations, we can derive two equations:
1. c - a = 1
2. c - 0 = 4
We can use the second equation to find the value of 'c':
c = 4
Now, substitute this value into the first equation to find 'a':
4 - a = 1
a = 3
Now, we can write the equation for the ellipse in standard form:
(x^2) / (a^2) + (y^2) / (b^2) = 1
Plug in the values of 'a' and 'b':
(x^2) / (3^2) + (y^2) / (4^2) = 1
Simplifying this equation gives us the standard form of an ellipse centered at the origin, with a vertex (0, 4), and a focus (0, 1):
(x^2) / 9 + (y^2) / 16 = 1
To find the equation of an ellipse in standard form, you need to know its center, the semi-major axis length, and the semi-minor axis length.
In this problem, the ellipse has its center at the origin (0, 0), and the given vertex is (0, 4). The relationship between the center and the vertex of an ellipse is that the center lies on the same vertical line as the vertex.
Since the center is at the origin (0, 0), and the vertex is along the same vertical line, the semi-major axis length (a) is 4.
Now, let's find the distance between the center and the focus, which gives us the value of c. The focus is given as (0, 1).
The distance between the center and the focus is given by the equation c = √(a^2 - b^2), where c is the distance between the center and the focus, a is the semi-major axis length, and b is the semi-minor axis length.
In this case, c = √(4^2 - b^2).
Since the focus is below the center, c = 4 - 1 = 3.
Now, you can use the given information about the center (0, 0), the value of a (4), and the value of c (3) to find the equation of the ellipse in standard form.
The equation of an ellipse centered at the origin in standard form is given by:
(x^2 / a^2) + (y^2 / b^2) = 1,
where a is the semi-major axis length and b is the semi-minor axis length.
Substituting the given values, we get the equation:
(x^2 / 4^2) + (y^2 / b^2) = 1.
Now, to find the value of b^2, you can use the relationship with c:
c^2 = a^2 - b^2,
where c is the distance between the center and the focus, a is the semi-major axis length, and b is the semi-minor axis length.
Substituting the given values, we get:
3^2 = 4^2 - b^2.
Simplifying the equation, we have:
9 = 16 - b^2.
Rearranging the equation, we get:
b^2 = 16 - 9 = 7.
Finally, substituting the value of b^2 in the equation of the ellipse, we get:
(x^2 / 4^2) + (y^2 / 7) = 1.
Thus, the equation in standard form of the given ellipse is:
(x^2 / 16) + (y^2 / 7) = 1.
center at (0,0) ... x^2/a^2 + y^2/b^2 = 1
c=1
a=4
b^2 = a^2-c^2 = 15
so,
x^2/16 + y^2/15 = 1