Find the Center 81x^2+36y^2+972x-648y+2916=0
The question requires you to find the coordinates of the center of an ellipse represented by the given quadratic equation 81x^2 + 36y^2 + 972x - 648y + 2916 = 0. To achieve this, the equation of the ellipse must be rewritten in standard form by completing the squares for both x and y terms, which allows for identifying the center, axes, and orientation of the ellipse.
Convert the given equation into the standard form of an ellipse.
Subtract
Complete the square for the
Identify coefficients for
Use the vertex form
Calculate
Insert
Simplify the fraction by canceling common factors.
Factor out a
Remove the common factor of
Simplify the fraction.
Reduce the fraction by canceling the common factor of
Determine
Plug in
Simplify the expression.
Perform the arithmetic operations.
Insert
Replace
Cancel out
Complete the square for the
Identify coefficients for
Use the vertex form
Calculate
Insert
Simplify the fraction by canceling common factors.
Factor out a
Remove the common factor of
Simplify the fraction.
Reduce the fraction by canceling the common factor of
Determine
Plug in
Simplify the expression.
Insert
Replace
Eliminate
Simplify the equation to get the standard form.
Recognize that the equation is now in the standard form of an ellipse.
Identify the center and axes lengths from the standard form.
The center of the ellipse is given by the coordinates
To solve for the center of an ellipse given in the general quadratic form, one must first rewrite the equation in the standard form of an ellipse. The standard form for an ellipse is:
The process involves:
Rearranging the equation to group the
Completing the square for both
Dividing through by the constant term to normalize the equation to equal
Comparing the resulting equation to the standard form to identify the center and axes lengths.
The vertex form of a parabola is used to complete the square, which is