Given an ellipse; 4x^2+y^2-48x-4y+48=0, find the Center, find the Major Axis, find the Minor Axis & the Distance from C to Foci?
To find the center, major axis, minor axis, and distance from the center to the foci of an ellipse given its equation, we need to rewrite the equation in a specific form called the standard form of an ellipse. The standard form equation for an ellipse with its center at (h, k), major axis length 2a, and minor axis length 2b is:
((x-h)^2 / a^2) + ((y-k)^2 / b^2) = 1
Comparing this form with the given equation, 4x^2 + y^2 - 48x - 4y + 48 = 0, we need to complete the square for both the x and y terms.
1. Complete the square for x terms:
Group the x terms together: (4x^2 - 48x)
Factor out a 4 from the x terms: 4(x^2 - 12x)
To complete the square, add and subtract (12/2)^2 = 36 to the expression inside the parentheses: 4(x^2 - 12x + 36 - 36)
Simplify: 4((x - 6)^2 - 36)
2. Complete the square for y terms:
Group the y terms together: (y^2 - 4y)
Factor out a 1 from the y terms: 1(y^2 - 4y)
To complete the square, add and subtract (4/2)^2 = 4 to the expression inside the parentheses: 1(y^2 - 4y + 4 - 4)
Simplify: 1((y - 2)^2 - 4)
Now, our equation becomes:
4((x - 6)^2 - 36) + 1((y - 2)^2 - 4) + 48 = 0
To simplify further, distribute the coefficients:
4(x - 6)^2 - 144 + (y - 2)^2 - 4 + 48 = 0
4(x - 6)^2 + (y - 2)^2 - 100 = 0
Divide the entire equation by -100 to put it in standard form:
((x - 6)^2 / 25) + ((y - 2)^2 / 100) = 1
Now we can compare this with the equation of a standard ellipse:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
From the standard form equation, we can determine the center, major axis, minor axis, and distance from the center to the foci:
1. Center: (h, k)
The center of the given ellipse is (6, 2).
2. Major Axis:
The major axis is determined by the term with the larger denominator. In this case, the major axis length is 2a, so the major axis length is 2 * sqrt(100) = 2 * 10 = 20.
3. Minor Axis:
The minor axis is determined by the term with the smaller denominator. In this case, the minor axis length is 2b, so the minor axis length is 2 * sqrt(25) = 2 * 5 = 10.
4. Distance from Center to Foci:
The distance from the center to the foci can be calculated using the formula: c = sqrt(a^2 - b^2), where a and b are the lengths of the major and minor axes, respectively. In this case, c = sqrt(100 - 25) = sqrt(75) = 5 * sqrt(3).
Therefore, the center of the given ellipse is (6, 2), the major axis length is 20, the minor axis length is 10, and the distance from the center to the foci is 5 * sqrt(3).