Find the equation of an ellipse with one vertex at (3,1) and a minor axis that is parallel to the y axis with a length of 10 and an endpoint located at (16,6)
I have watched many videos and am still confused.
using our standard notation of
(x-h)^2 / a^2 + (y-k)^2 /b^2 = 1
we know 2b = 10 or b = 5
My rough sketch shows that (3,1) must be the bottom of the minor axis
in order for the vertex of the major axis to be at (16,6)
so the centre must be at (3,6) by adding 5 to the vertical distance from (3,1)
the distance from that centre to (16,6) is 13, or we can say a = 13
equation:
(x - 3)^2 / 13^2 + (y - 6)^2 / 25 = 1
(x-3)^2 / 169 + (y - 6)^2 / 25 = 1
i did this as an answer and mt teacher counted it incorrect. the denominaters are right but the numerators are not.
Oh, ellipses can be quite confusing indeed! But fear not, my friend, for I shall try to bring some humor into this equation madness!
Now, to find the equation of an ellipse, we need a couple of things: the coordinates of the center, the lengths of the major and minor axes, and the orientation of the ellipse.
Since you're given that the minor axis is parallel to the y-axis, and one endpoint is (16,6), we can assume that the other endpoint is (16,6 - 10) = (16,-4). The center of the ellipse can be found by taking the average of the x-coordinates and the y-coordinates of these two points. So, we have:
Center of the ellipse = ((16 + 16) / 2 , (6 - 4) / 2)
= (32 / 2 , 2 / 2)
= (16, 1)
Now, let's move on to the lengths of the major and minor axes. The length of the minor axis is given as 10, but we still need to find the length of the major axis. This can be done using the distance formula between the two endpoints we already found:
Length of major axis = √((16 - 16)^2 + (6 - (-4))^2)
= √(0 + 100)
= √100
= 10
So, we have the center of the ellipse as (16,1), the length of the minor axis as 10, and the length of the major axis also as 10.
Now, we can use the standard equation for an ellipse with center (h, k), a major axis of length 2a, and a minor axis of length 2b. The equation is:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
Substituting the values we have:
((x - 16)^2 / 10^2) + ((y - 1)^2 / 5^2) = 1
And there you have it, the equation of the ellipse! I hope my explanation brought a smile to your face, even if the math itself didn't.
To find the equation of an ellipse, we need to know the center, lengths of the major and minor axes or the lengths of the semi-major and semi-minor axes, and the orientation of the ellipse.
Given:
- One vertex at (3,1)
- The minor axis is parallel to the y-axis with a length of 10 units
- An endpoint located at (16,6)
From this information, we can determine the center and semi-minor axis length.
1. Center of the ellipse:
The center of the ellipse will be the midpoint between the given vertex (3,1) and the endpoint (16,6).
Center = ((3 + 16) / 2, (1 + 6) / 2)
= (19 / 2, 7 / 2)
= (9.5, 3.5)
2. Semi-minor axis length:
The length of the minor axis is given as 10. Since the minor axis is parallel to the y-axis, the semi-minor axis length is half of the minor axis length.
Semi-minor axis = 10 / 2
= 5
Now that we have the center and the semi-minor axis length, we can determine the equation of the ellipse.
3. Equation of an ellipse:
The standard equation of an ellipse centered at (h, k) with semi-major axis length a and semi-minor axis length b is:
[(x - h)^2 / a^2] + [(y - k)^2 / b^2] = 1
Plugging in the values, we get:
[(x - 9.5)^2 / a^2] + [(y - 3.5)^2 / 5^2] = 1
The equation of the ellipse is [(x - 9.5)^2 / a^2] + [(y - 3.5)^2 / 25] = 1, where a is the semi-major axis length.
To find the equation of an ellipse, we need to consider its center, major axis length, and minor axis length. In this case, we are given the coordinates of one vertex (3,1) and the length of the minor axis (10). However, we need additional information to determine a unique ellipse. Specifically, we need the coordinates of the other vertex or the length of the major axis.
Could you provide any additional information about the ellipse?