find the equation for an ellipse that satisfies the following condition
Vertices at (-3,1) and (9,1)
one focus at (7,1)
I will assume you know the general equation of an ellipse and its properties in terms of a,b and c.
the centre would be the midpoint of (-3,1) and (9,1) which is (3,1) from which we can easily see that a = 6
one focal point is (7,1) so the distance from the centre to the focal point is 4
therefore c=4
now in a horizontally placed ellipse
b^2 + c^2 = a^2
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Your equation should be
(x-3)^2/ 36 + (y-1)^2 /48 = 1
To find the equation for an ellipse that satisfies the given conditions, follow these steps:
1. Find the center of the ellipse: The center of the ellipse is the midpoint between the vertices (-3,1) and (9,1). Use the midpoint formula to find the x-coordinate and y-coordinate of the center. The midpoint formula is (x1 + x2)/2 and (y1 + y2)/2. In this case, the x-coordinate of the center is (-3 + 9)/2 = 3 and the y-coordinate of the center is (1 + 1)/2 = 1. So, the center of the ellipse is (3,1).
2. Find the distance from the center to one of the foci: Since one of the foci is given as (7,1), you can find the distance from the center (3,1) to that focus using the distance formula. The distance formula is √((x2 - x1)^2 + (y2 - y1)^2). In this case, the distance is √((7 - 3)^2 + (1 - 1)^2) = √(16 + 0) = √16 = 4.
3. Determine the value of c: In an ellipse, the distance from the center to each focus is denoted as c. So, in this case, c = 4.
4. Determine the value of a: In an ellipse, a is the semi-major axis, which is half the distance between the vertices. The vertices are given as (-3,1) and (9,1). The distance between these two points is |9 - (-3)| = 12. Therefore, a = 12/2 = 6.
5. Use the formula for an ellipse: The equation for an ellipse in standard form is (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h,k) is the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis.
Plugging in the values we found:
(x - 3)^2 / 6^2 + (y - 1)^2 / b^2 = 1
6. Solve for b: To find b, we can use the relationship between a, b, and c in an ellipse: b^2 = a^2 - c^2. Plugging in the values, b^2 = 6^2 - 4^2 = 36 - 16 = 20. Therefore, b = √20 = 2√5.
7. Final equation: Now we can substitute the values of a, b, and the center (h,k) into the equation to get the final equation for the ellipse. The equation becomes:
(x - 3)^2 / 6^2 + (y - 1)^2 / (2√5)^2 = 1
Simplifying:
(x - 3)^2 / 36 + (y - 1)^2 / 20 = 1
So, the equation for the ellipse that satisfies the given conditions is (x - 3)^2 / 36 + (y - 1)^2 / 20 = 1.