Find the equation of ellipse whose center Is the origin one focus at (2,2) and length of the semi minor axis is √8
so you have
c = 2√2
b = 2√2
so a = 4
If the major axis were along the x-axis, then the equation would be
x^2/16 + y^2/8 = 1
But you have rotated it by 45° so you need to apply the rotation matrix to obtain the new x' and y' equation
since the centre is (0,0) and the major axis passes through (2,2), you are
looking at an ellipse which has been rotated 45° from its standard position.
So in that standard position, c = √(2^2 + 2^2) = √8
but the semi minor axis is also √8, and would not change for any rotation
so we have a^2 = b^2 + c^2 = 8+8 = 16 , (from b = √8)
and a = 4
so the ellipse before its rotation was x^2/16 + y^2/8 = 1
8x^2 + 16y^2 = 128
Now apply the rotation matrix to this equation:
x = u cosθ + vsinθ = u√2/2 + v√2/2 , since θ = 45°
y = -usinθ + vcosθ = -u√2/2 + v√2/2
where u and v will be the x and y of our rotated equation
8(u√2/2 + v√2/2)^2 + 16(-u√2/2 + v√2/2)^2 = 128
8(u^2/2 + uv + v^2/2) + 16(u^2/2 - uv + v^2/2) = 128
4u^2 + 8uv + 4v^2 + 8u^2 - 16uv + 8v^2 = 128
12u^2 - 8uv + 12v^2 = 128
3u^2 - 2uv + 3v^2 = 32
or, in terms of x and y:
3x^2 - 2xy + 3y^2 = 32
whewww!, I graphed it on Desmos and it is correct, yeahhh
To find the equation of the ellipse, we need to determine the lengths of the semi-major axis (a) and semi-minor axis (b) first.
1. Given that one focus is at (2,2) and the center is at the origin, the distance between the center and one focus is the value of c. We can use the distance formula to find c:
c = √[(2-0)^2 + (2-0)^2]
= √(2^2 + 2^2)
= √(4 + 4)
= √8
Therefore, c = √8.
2. The length of the semi-minor axis is given as √8. This is the distance between the origin and the co-vertex, which is perpendicular to the major axis.
So, b = √8.
3. To find the length of the semi-major axis (a), we know that c^2 = a^2 - b^2. Substituting the values we have:
(√8)^2 = a^2 - (√8)^2
8 = a^2 - 8
a^2 = 8 + 8
a^2 = 16
a = √16
a = 4
Therefore, a = 4.
4. Finally, we can write the equation of the ellipse using the standard form:
(x^2 / a^2) + (y^2 / b^2) = 1
Plugging in the values we found, the equation of the ellipse is:
(x^2 / 4^2) + (y^2 / (√8)^2) = 1
(x^2 / 16) + (y^2 / 8) = 1
Simplifying, the equation of the ellipse is:
(x^2 / 16) + (y^2 / 8) = 1
To find the equation of an ellipse, we need to know the coordinates of the center, the lengths of the semi-major and semi-minor axes, and the position of one focus. In this case, we are given the center (0,0), one focus (2,2), and the length of the semi-minor axis (√8).
Let's denote the semi-major axis as a and the semi-minor axis as b.
The distance between the center and one focus is given by c, where c represents the distance from the center to either of the foci. In this case, the distance between the center (0,0) and the focus (2,2) is √(2^2 + 2^2) = √8.
We can calculate the value of a using the equation: a = √(b^2 + c^2).
Given that b = √8, we have:
a = √(8 + 8) = √16 = 4.
Now that we have the values of a and b, we can write the equation of the ellipse:
(x^2 / a^2) + (y^2 / b^2) = 1.
Substituting the values of a and b, we get:
(x^2 / 4^2) + (y^2 / (√8)^2) = 1.
Simplifying further:
(x^2 / 16) + (y^2 / 8) = 1.
Therefore, the equation of the ellipse whose center is the origin, one focus is at (2,2), and the length of the semi-minor axis is √8 is:
x^2/16 + y^2/8 = 1.