I need help to graph the circle
4x^2+4y^2+16x-32y+71=0
Graph the circle
To graph the circle given by the equation 4x^2+4y^2+16x-32y+71=0, we can follow these steps:
Step 1: Rewrite the equation in the standard form
To graph a circle, we want the equation to be in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle, and r represents its radius.
We can begin by completing the square:
4x^2+4y^2+16x-32y+71=0
Divide the entire equation by 4 to simplify:
x^2 + y^2 + 4x - 8y + 17.75 = 0
Rearranging the terms:
x^2 + 4x + y^2 - 8y = -17.75
Completing the square for the x-terms:
(x^2 + 4x + 4) + y^2 - 8y = -17.75 + 4
Completing the square for the y-terms:
(x^2 + 4x + 4) + (y^2 - 8y + 16) = -17.75 + 4 + 16
Simplifying:
(x + 2)^2 + (y - 4)^2 = 2.25
So, the equation of the given circle can be written in standard form as (x + 2)^2 + (y - 4)^2 = 2.25.
Step 2: Identify the center and radius
From the standard form equation, we can identify the center of the circle as the point (-2, 4) and the radius as the square root of 2.25, which is 1.5.
Step 3: Plot the center and radius on a graph
On a Cartesian plane, plot the center (-2, 4).
Step 4: Draw the circle using the center and radius
Using the center (-2, 4) and the radius of 1.5, draw a circle passing through the point (-2, 4).
This circle represents the graph of the equation 4x^2+4y^2+16x-32y+71=0.