Determine equation of the ellipse if the distance between its foci is 2sqrt5 and the end points of its minor axis are (-1,-2) and (3,-2).
To determine the equation of an ellipse, we need to find the center and the lengths of the major and minor axes. Given that the distance between the foci is 2√5 and the endpoints of the minor axis are (-1,-2) and (3,-2), we can start by finding the center of the ellipse.
Step 1: Find the center of the ellipse
The center of the ellipse is the midpoint of the minor axis. To find the midpoint, we can use the formula:
Midpoint = ( (x1 + x2) / 2, (y1 + y2) / 2)
Given points:
(x1, y1) = (-1, -2)
(x2, y2) = (3, -2)
Midpoint = ( ((-1) + 3) / 2, ((-2) + (-2)) / 2)
= (2 / 2, (-4) / 2)
= (1, -2)
So, the center of the ellipse is (1, -2).
Step 2: Find the length of the major axis
The major axis is the distance between the two endpoints of the minor axis. In this case, the endpoints are (-1,-2) and (3,-2), and since they are on the same horizontal line, the length of the major axis is equal to the distance between their x-coordinates.
Length of major axis = |x2 - x1|
= |3 - (-1)|
= |4|
= 4
So, the length of the major axis is 4.
Step 3: Find the length of the minor axis
The length of the minor axis is equal to 2 times the distance between the foci. In this case, the distance between the foci is given as 2√5, so the length of the minor axis is 2 times that.
Length of minor axis = 2 * 2√5
= 4√5
So, the length of the minor axis is 4√5.
Step 4: Write the equation of the ellipse
The equation of an ellipse with center (h, k), major axis length 2a, and minor axis length 2b is:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Substituting the values we found:
Center (h, k) = (1, -2)
Major axis length a = 4 / 2 = 2
Minor axis length b = 4√5 / 2 = 2√5
Equation of the ellipse:
(x - 1)^2 / 2^2 + (y + 2)^2 / (2√5)^2 = 1
Simplifying:
(x - 1)^2 / 4 + (y + 2)^2 / 20 = 1
Therefore, the equation of the ellipse is:
(x - 1)^2 / 4 + (y + 2)^2 / 20 = 1
To determine the equation of an ellipse, we typically need to know the coordinates of its center, the lengths of its major and minor axes, and the orientation of the ellipse. However, in this case, we are given the distance between the foci and the end points of the minor axis. We can work with this information to determine the equation.
Let's start by finding the coordinates of the center of the ellipse. Since the end points of the minor axis are (-1, -2) and (3, -2), the center lies on the line y = -2, which is the equation of the minor axis. This means that the y-coordinate of the center is -2.
Next, we need to find the x-coordinate of the center. The distance between the foci is given as 2√5, which means the distance between the center and each focus is √5. Since the foci are on the major axis, their y-coordinates are the same as the center, which is -2.
To determine the x-coordinate of the center, we can use the Pythagorean theorem. Let's consider one half of the distance between the foci, which is √5. This half distance forms a right triangle with the x-coordinate of the center as one leg and the distance from the center to one of the foci as the hypotenuse.
Using the Pythagorean theorem, we have:
(x - a)^2 + (y - b)^2 = c^2
Plugging in the values we have, we get:
(x - a)^2 + (-2 - (-2))^2 = (√5)^2
(x - a)^2 + 0 = 5
(x - a)^2 = 5
Since we only have one term in the equation, we know that the ellipse is centered on the y-axis (a vertical ellipse) and the x-coordinate of the center, a, is 0.
Therefore, the equation of the ellipse is:
(x - 0)^2/5 + (y + 2)^2/b^2 = 1
To find the value of b, we need to determine the length of the semi-minor axis. The distance between the end points of the minor axis is 4 units, so the length of the semi-minor axis is 2 units. Plugging this value into the equation, we get:
(x)^2/5 + (y + 2)^2/2^2 = 1
Simplifying further, we have:
(x^2)/5 + (y + 2)^2/4 = 1
Therefore, the equation of the ellipse is:
(x^2)/5 + (y + 2)^2/4 = 1
the minor has has length 2b, so
b = 4/2 = 2
c = √5
so a^2 = b^2 + c^2 = 9
The center is midway between the foci, at (1,-2)
The minor axis is horizontal, so the major axis is vertical, making the equation
(x-1)^2/4 + (y+2)^2/9 = 1
Confirm at
https://www.wolframalpha.com/input/?i=%28x-1%29%5E2%2F4+%2B+%28y%2B2%29%5E2%2F9+%3D+1
and click on Properties
Why did the ellipse go to the doctor? Because it had an axis disorder!
Now, let's find the equation of the ellipse. We know that the distance between its foci is 2√5, so let's call that distance 2c. That means c = √5.
We also have the endpoints of the minor axis, which are (-1, -2) and (3, -2). The length of the minor axis, which we can call 2b, is simply the distance between these two points. So, 2b = 3 - (-1) = 4. That means b = 2.
Now, we can find a using the relationship: c^2 = a^2 - b^2. Plugging in the values we have: (√5)^2 = a^2 - 2^2. Simplifying this, 5 = a^2 - 4, which gives us a^2 = 9, or a = ±3.
Since the given endpoints lie on the minor axis, the equation of the ellipse centered at the origin (0, 0) can be written as:
(x^2/9) + (y^2/4) = 1.
So, the equation of the ellipse is (x^2/9) + (y^2/4) = 1. And remember, if the equation of the ellipse makes you feel a bit off-center, just remember that humor can always bring you back to the right place!