Hi how would i solve this??
The equation of the cirlce is
x^2+y^2-4x+2y=0. Determine the coordinates of the center and the radius of the circle.
What do i do?
anyone
To determine the center and radius of the circle represented by the equation x^2 + y^2 - 4x + 2y = 0, you can follow a few steps:
1. Rearrange the equation: Start by rearranging the equation so that it is in a standard form for the equation of a circle, which is (x - h)^2 + (y - k)^2 = r^2.
In this case, rearranging the equation gives us (x^2 - 4x) + (y^2 + 2y) = 0.
2. Complete the square: By completing the square, you can determine the values of h, k, and r, which correspond to the center and radius of the circle.
For x^2 - 4x, take half of the coefficient of x (-4), which is -2. Square that value (-2^2 = 4), and then add it to both sides of the equation. This is done to maintain equality.
(x^2 - 4x + 4) + (y^2 + 2y) = 4
Similarly, for y^2 + 2y, take half of the coefficient of y (2), which is 1. Square that value (1^2 = 1), and then add it to both sides of the equation.
(x^2 - 4x + 4) + (y^2 + 2y + 1) = 4 + 1
Simplify the equation:
(x - 2)^2 + (y + 1)^2 = 5
3. Compare to standard form: By comparing the rearranged equation to the standard form, (x - h)^2 + (y - k)^2 = r^2, you can identify the values of h, k, and r.
From the simplified equation, we can determine that the center of the circle is at the point (2, -1), because (h, k) represents the coordinates of the center.
Furthermore, since r^2 = 5, we can calculate the radius, r, by taking the square root of both sides. The positive square root of 5 yields the radius of the circle.
Therefore, the center of the circle is at (2, -1), and the radius is √5 (approximately 2.236).