Find the vertices, center, asymptotes, and foci of the ellipse and sketch its graph: 9x^2-4y^2-18x+8y+41=0
9x^2-4y^2-18x+8y+41=0
9(x^2 - 2x + .....) - 4(y^2 - 2y + ....) = -41
9(x^2 - 2x + 1) - 4(y^2 - 2y + 1) = -41 + 1 + 1
I will let you finish it from there.
You do realize of course, that your equation does NOT represent an ellipse.
You forgot to multiply the 9 and the 4 to make the equation = -41 + 9 - 4
you are right, I messed up
To find the vertices, center, asymptotes, and foci of the ellipse, we will need to rewrite the given equation in a standard form. The standard form of the equation of an ellipse is:
((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) = 1, where (h, k) represents the center of the ellipse, a represents the semi-major axis, and b represents the semi-minor axis.
Let's rewrite the given equation accordingly:
9x^2 - 4y^2 - 18x + 8y + 41 = 0
Rearranging the terms, we get:
9x^2 - 18x - 4y^2 + 8y = -41
We can complete the square separately for x and y terms. Let's start with x:
9(x^2 - 2x) - 4y^2 + 8y = -41
To complete the square for x, we need to divide the coefficient of x by 2, square it, and add and subtract it inside the parentheses:
9(x^2 - 2x + 1) - 1 - 4y^2 + 8y = -41
Simplifying further:
9(x - 1)^2 - 4y^2 + 8y - 1 = -41
-4y^2 + 8y - 41 = - 9(x - 1)^2
Next, let's complete the square for y by dividing the coefficient of y by 2, squaring it, and adding and subtracting it:
-4(y^2 - 2y) - 41 = -9(x - 1)^2
-4(y^2 - 2y + 1) + 4 - 41 = -9(x - 1)^2
-4(y - 1)^2 - 37 = -9(x - 1)^2
Rearranging the terms to match the standard form, we have:
-9(x - 1)^2 - 4(y - 1)^2 = 37
Dividing both sides by 37, we get:
(-9(x - 1)^2) / 37 - (4(y - 1)^2) / 37 = 1
Now, comparing this equation with the standard form:
((x - h)^2) / a^2 + ((y - k)^2) / b^2 = 1
We can see that the center of the ellipse is (h, k) = (1, 1).
To find the semi-major axis (a) and semi-minor axis (b), we take the square root of the denominators:
a = sqrt(37/9)
b = sqrt(37/4)
Therefore, the semi-major axis (a) is approximately 1.53 and the semi-minor axis (b) is approximately 1.92.
The vertices of the ellipse can be found by adding or subtracting the semi-major axis from the center:
Vertex 1: (1 - a, 1)
Vertex 2: (1 + a, 1)
The vertices are approximately (0.53, 1) and (1.47, 1).
To find the foci, we can use the formula:
c = sqrt(a^2 - b^2)
Foci 1: (1 - c, 1)
Foci 2: (1 + c, 1)
The foci are approximately (0.66, 1) and (1.34, 1).
To find the asymptotes, we can use the formulas:
y = k ± (b/a)(x - h)
The equations of the asymptotes are:
y = 1 ± (1.92/1.53)(x - 1)
Simplifying further:
y = 1 ± (1.26)(x - 1)
Now we can sketch the graph of the ellipse using the center, vertices, foci, and asymptotes as reference points.