Write the standard equation of the ellipse with the given characteristics
Vertices: (0,7) (0,-7)
foci: (0,2) (0,-2)
If this is Ajay , please don't switch names
this is a routine question
looks like (0,0) is the centre
a = 7 , and c = 2
we know
b^2 + c^2= a^2
b^2+ 4 = 49
b^2= 45
x^2 /49 + y^2 / 45 = 1
To write the standard equation of an ellipse, we need to use the distance formula and some properties of ellipses.
First, let's label the important points on the ellipse:
Center = (h, k)
Vertices = (0, 7) and (0, -7)
Foci = (0, 2) and (0, -2)
The distance formula to find the distance between two points (x1, y1) and (x2, y2) is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Now, let's calculate the distance from the center to one of the foci:
d = √((0 - 0)^2 + (2 - k)^2)
Since the distance to the foci is given as 2, we can set up the following equation:
√((0 - 0)^2 + (2 - k)^2) = 2
Simplifying the equation:
(2 - k)^2 = 4
Expanding:
4 - 4k + k^2 = 4
Moving terms to one side:
k^2 - 4k = 0
Now, let's calculate the distance from the center to one of the vertices:
d = √((0 - 0)^2 + (7 - k)^2)
Since the distance to the vertices is given as 7, we can set up the following equation:
√((0 - 0)^2 + (7 - k)^2) = 7
Simplifying the equation:
(7 - k)^2 = 49
Expanding:
49 - 14k + k^2 = 49
Moving terms to one side:
k^2 - 14k = 0
From the two equations, we can see that k = 0 or k = 14. However, since the foci are closer to the center than the vertices, we can conclude that k = 0.
Now we know that the center of the ellipse is (h, k) = (0, 0).
Next, let's calculate the distance from the center to one of the foci, which is given as 2. We can use the equation:
c^2 = a^2 - b^2
where c is the distance from the center to a focus and a and b are the semi-major and semi-minor axes, respectively.
Given the foci distance c = 2 and the vertices distance a = 7, we have:
2^2 = 7^2 - b^2
4 = 49 - b^2
b^2 = 49 - 4
b^2 = 45
Now we have the values of a^2 = 49 and b^2 = 45.
By using the standard equation formula for an ellipse with a horizontal major axis, we can write:
x^2/a^2 + y^2/b^2 = 1
Substituting the values, we get:
x^2/49 + y^2/45 = 1
Therefore, the standard equation of the ellipse is:
x^2/49 + y^2/45 = 1