You must transform the equation in order to identify the center point and the radii in order to sketch the graph.

Transformation involves completing the square as for a circle with a couple of extra steps.

### Step-by-Step Example

 4x2 + 9y2 – 16x + 90y + 205 = 0 Given this equation in standard form whereA = 4, B = 0, C = 9, D = -16, E = 10, F = 205 4x2 – 16x + 9y2 + 90y = -205 Rearrange the terms, group like terms and move the constant to the opposite side of the equation. 4(x2 – 4x + ___) + 9(y2 + 10y + ___) = -205 Group “x” terms and “y” terms and factor out the coefficients of the squared terms. 4(x2 – 4x + 4) + 9(y2 + 10y + 25) = -205 + (4 x 4) + (9 x 25) Complete each of the “squares” by adding the appropriate quantities and add like quantities to the opposite side of the equation. 4(x – 2)2 + 9(y + 5)2 = 36 Write perfect squares in factored form and combine the constants four times x minus two squared over thirty-six 4(x − 2)2 36 + nine times y plus five squared over thirty-six 9y + 5)2 36 = thirty-six over thirty-six 36 36 Divide each term by 36 so the left side equals 1. x minus two squared over nine (x − 2)2 9 + y plus five squared over four (y + 5)2 4 = 1 Simplify all terms!

Use your own graph paper or go to Free Graph Paper (this will open a new window/tab in your browser. Close the window/tab to return to this lesson) and sketch the graph.

h = __________ k = __________ center point = __________

Interactive popup. Assistance may be required.

h = 2, k = -5 center = (2, -5) Interactive popup. Assistance may be required.

Now try to transform, if needed, and graph the following:

1. 16x2 + 4y2 + 96x – 32y + 144 = 0

Interactive popup. Assistance may be required.  2. x squared over forty-nine x2 49 + y minus eight squared over one hundred (y - 8)2 100 = 1

Interactive popup. Assistance may be required.

No need to transform.
Center = (0, 8), rx = 7, and ry = 10 