write the equation of the ellipse that has its centre at the origin with focus (0,4) and vertex at (0,7)

the major axis lies along the y-axis, so

a^2 + 4^2 = 7^2
a^2 = 33
a=√33

x^2/33 + y^2/49 = 1

please help me i don't understand

Why did the ellipse go to therapy? Because it felt so centered, but had a focus on everything!

Now, let's break down the equation of this ellipse.

Given that the center is at the origin (0,0), the standard form of the equation for an ellipse is:

(x^2/a^2) + (y^2/b^2) = 1

Now, the distance from the center to the vertex is 7, which gives us the value of "a", and the distance from the center to the focus is 4, giving us the value of "c".

Using the relationship between "a", "b", and "c" in an ellipse, we have the equation:

c^2 = a^2 - b^2

Substituting the given values, we have:

4^2 = 7^2 - b^2
16 = 49 - b^2
b^2 = 49 - 16
b^2 = 33

So, the equation of the ellipse is:

(x^2/49) + (y^2/33) = 1

But remember, it's important to take every equation with a grain of humor - and that's the joke!

To find the equation of an ellipse, we need to determine the values of a, b, and c, where "a" represents the semi-major axis, "b" represents the semi-minor axis, and "c" represents the distance between the center and one of the foci.

We are given that the center of the ellipse is at the origin, which means the coordinates of the center are (0,0). The coordinates of the focus are (0,4), and the coordinates of the vertex are (0,7).

First, let's find the value of "c" using the distance formula:

c = distance between center and focus
= √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(0 - 0)^2 + (4 - 0)^2]
= √(0 + 16)
= √16
= 4

We know that the distance from the center to one of the vertices is equal to "ae," where "e" represents the eccentricity. In this case, the distance is 7 units. Since the center is at the origin, "ae = 7," which means "a = 7/e."

Now, we can substitute the value of "c" into the equation "c = ae" to solve for the eccentricity "e":

4 = (7/e) * e
4e = 7
e = 7/4

We can also determine the value of "b" from the distance from the center to one of the co-vertices:

b = √(a^2 - c^2)
= √((7/e)^2 - 4^2)
= √(49/e^2 - 16)

Therefore, the equation of the ellipse is:

x^2 / (7/e)^2 + y^2 / (√(49/e^2 - 16))^2 = 1

To write the equation of an ellipse that has its center at the origin, focus at (0,4), and vertex at (0,7), we need to determine the major and minor axes lengths.

The major axis length is equal to twice the distance between the center and the vertex, which in this case is 2 * 7 = 14.

The minor axis length is equal to twice the distance between the center and each focus point, which in this case is 2 * 4 = 8.

Since the ellipse is centered at the origin, the equation of the ellipse can be written as:

(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, since the major axis length is 14, the semi-major axis length is a = 14/2 = 7.

And since the minor axis length is 8, the semi-minor axis length is b = 8/2 = 4.

Thus, the equation of the ellipse is:

(x^2 / 7^2) + (y^2 / 4^2) = 1.

Simplifying, we get:

(x^2 / 49) + (y^2 / 16) = 1.

Therefore, the equation of the ellipse is (x^2 / 49) + (y^2 / 16) = 1.