How do I write an equation for the ellipse with the following: Foci (9,0) &(-9,0) and x-intercepts (15,0) and (-15,0)
Well, let's start with a joke to lighten the mood:
Why did the ellipse go to the gym?
Because it wanted to get in shape!
Now, let's get to business. To write the equation for an ellipse, we need to know the coordinates of the foci and the lengths of the major and minor axes. Lucky for us, we have all the information we need!
The distance from the center of the ellipse to each focus is known as the focal length (denoted by the letter "c"). In this case, since the foci are at (9, 0) and (-9, 0), we can determine that c = 9.
The distance from the center to each x-intercept is half of the length of the major axis (denoted by "a"). Since the x-intercepts are at (15, 0) and (-15, 0), we know that a = 15.
Now, let's put it all together. The equation for an ellipse centered at the origin (0, 0) with foci (c, 0) and (–c, 0) and x-intercepts (a, 0) and (-a, 0) can be written as:
((x - h)^2/a^2) + ((y - k)^2/b^2) = 1
Since our ellipse is centered at the origin, (h, k) = (0, 0). Plugging in the values we know, we get:
(x^2/225) + (y^2/b^2) = 1
Now let's find the value of b. We can use the relationship between a, b, and c:
c^2 = a^2 - b^2
Since c = 9 and a = 15, we can solve for b:
9^2 = 15^2 - b^2
81 = 225 - b^2
b^2 = 144
Taking the square root of both sides, we find that b = 12.
So the equation for the ellipse is:
(x^2/225) + (y^2/144) = 1
Hope that helps! And remember, even math problems can have funny solutions!
To write the equation for an ellipse with the given foci and x-intercepts, you can follow these steps:
Step 1: Find the center of the ellipse.
The center of the ellipse is the midpoint between the two foci:
Center = ( (9 + (-9))/2 , (0 + 0)/2 )
Center = (0, 0)
Step 2: Find the distance between each focus and the center.
The distance between each focus and the center is half the distance between the x-intercepts:
Distance = (15 - 0)/2 = 7.5
Step 3: Find the value of "a".
The value of "a" represents the distance from the center to each vertex, which is equal to the distance between each focus and the center:
a = 7.5
Step 4: Find the value of "b".
The value of "b" represents the distance from the center to each co-vertex, which can be found using the Pythagorean theorem. The distance between the center and each co-vertex is the square root of the difference between the squares of the distance between the center and one of the x-intercepts and the distance "a" squared:
b = √(15^2 - 7.5^2)
b = 13.39 (approximately)
Step 5: Determine the major and minor axes.
Since a > b in this case, the major axis is the x-axis, and the minor axis is the y-axis.
Step 6: Write the equation of the ellipse.
The standard equation for an ellipse with the center at the origin is:
(x^2 / a^2) + (y^2 / b^2) = 1
Substituting the values of "a" and "b" obtained from the previous steps:
(x^2 / 7.5^2) + (y^2 / 13.39^2) = 1
Therefore, the equation of the ellipse with the given foci (9,0) and (-9,0), and x-intercepts (15,0) and (-15,0) is:
(x^2 / 56.25) + (y^2 / 179.2921) = 1
To write the equation of an ellipse given its foci and x-intercepts, we need to find the necessary information such as the center and semi-major and semi-minor axes. Follow the steps below:
Step 1: Find the coordinates of the center:
The center of the ellipse is the midpoint between the two foci. Given the foci (9, 0) and (-9, 0), the x-coordinate of the center is the average of the x-coordinates of the foci, which is (9 + -9)/2 = 0. The y-coordinate remains the same, so the center of the ellipse is (0, 0).
Step 2: Find the lengths of the semi-major and semi-minor axes:
The semi-major axis of an ellipse is half the distance between the two x-intercepts. Given the x-intercepts (15, 0) and (-15, 0), the semi-major axis length is (15 - (-15))/2 = 30/2 = 15.
The semi-minor axis of an ellipse is half the distance between the center and one of the y-intercepts. Since the y-intercepts are both 0, the semi-minor axis length is 0.
Step 3: Write the equation of the ellipse:
The equation of an ellipse with center (h, k), semi-major axis length a, and semi-minor axis length b is given by:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Plugging in the values we found:
- Center (h, k): (0, 0)
- Semi-major axis length a: 15
- Semi-minor axis length b: 0
The equation of the ellipse becomes:
(x - 0)^2 / 15^2 + (y - 0)^2 / 0^2 = 1
Simplifying the equation:
x^2 / 15^2 + y^2 / 0 = 1
Since the semi-minor axis length is 0, the equation becomes:
x^2 / 15^2 = 1
Therefore, the equation of the ellipse is:
x^2 / 225 = 1