given an ellipse; 2x^2+6y^2+32x-48y+212=0, find center, major axis, minor axis & distance from c to foci?
To find the center, major axis, minor axis, and distance from the center to the foci of the ellipse given by the equation 2x^2 + 6y^2 + 32x - 48y + 212 = 0, we can follow a step-by-step approach:
Step 1: Rewrite the equation of the ellipse in standard form: (x - h)^2/a^2 + (y - k)^2/b^2 = 1.
Step 2: Group the x-terms and y-terms together by completing the square for each variable.
Starting with the x-terms:
2x^2 + 32x = -6y^2 + 48y - 212.
Divide the equation by 2 to simplify:
x^2 + 16x = -3y^2 + 24y - 106.
To complete the square for x, we take half the coefficient of the x-term (which is 16), square it (which is 256), and add it to both sides of the equation:
x^2 + 16x + 256 = -3y^2 + 24y - 106 + 256.
This yields:
(x + 8)^2 = -3y^2 + 24y + 150.
Perform the same completion of the square process for the y-terms:
-3y^2 + 24y = (x + 8)^2 - 150.
Divide the equation by -3 to simplify:
y^2 - 8y = -(1/3)(x + 8)^2 + 50.
To complete the square for y, we take half the coefficient of the y-term (which is -8), square it (which is 16), and add it to both sides of the equation:
y^2 - 8y + 16 = -(1/3)(x + 8)^2 + 50 + 16.
This yields:
(y - 4)^2 = -(1/3)(x + 8)^2 + 66.
Step 3: Rewrite the equation to match the standard form:
((x + 8)^2) / (33) + ((y - 4)^2) / (66) = 1.
From the standard form, we can identify the center of the ellipse, which is (-8, 4).
The major axis of an ellipse is the longer axis, which is determined by the larger denominator in the standard form equation. In this case, the major axis is along the y-axis (vertical) with length 2sqrt(66).
The minor axis of an ellipse is the shorter axis, which is determined by the smaller denominator in the standard form equation. In this case, the minor axis is along the x-axis (horizontal) with length 2sqrt(33).
Step 4: Calculate the distance from the center to the foci.
The distance from the center to the foci can be determined using the formula c = sqrt(a^2 - b^2), where c is the distance from the center to the foci and a and b are the semi-major and semi-minor axes of the ellipse, respectively.
For our ellipse, a = sqrt(66) and b = sqrt(33). Substituting these values into the formula:
c = sqrt(66 - 33) = sqrt(33).
Therefore, the distance from the center to the foci is sqrt(33).