x^2+y^2+x-6y+9=0
Graph the circle using the center (h,k) and radius r. find the intercepts
first, complete the squares:
x^2+x + y^2-6y = -9
x^2+x+(1/4) + y^2-6y+9 = -9 + 1/4 + 9
(x + 1/2)^2 + (y-3)^2 = 1/4
Now all should be clear. See
http://www.wolframalpha.com/input/?i=x^2%2By^2%2Bx-6y%2B9%3D0
To graph the circle given by the equation x^2+y^2+x-6y+9=0, we need to rearrange the equation in the standard form of a circle, which is (x-h)^2 + (y-k)^2 = r^2. In this form, (h, k) represents the coordinates of the center, and r represents the radius.
Let's start by completing the square for the x and y terms:
x^2 + x + y^2 - 6y + 9 = 0
Rearrange the equation by grouping the x terms and the y terms:
(x^2 + x) + (y^2 - 6y) + 9 = 0
Now, complete the square within parentheses for both x and y terms:
(x^2 + x + 1/4) + (y^2 - 6y + 9) + 9 = 1/4 + 9 + 9
Factor the completed square terms:
(x + 1/2)^2 + (y - 3)^2 + 9 = 19 + 9
Simplify:
(x + 1/2)^2 + (y - 3)^2 = 37
Now the equation is in the standard form (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. So, we can identify that the center of the circle is (-1/2, 3), and the radius is the square root of 37.
To find the intercepts, we substitute 0 for both x and y in the circle equation and solve for the remaining variable.
For the x-intercepts (when y = 0):
(-1/2)^2 + (0 - 3)^2 = 37
1/4 + 9 = 37
37.25 = 37
Since the equation is not true, there are no x-intercepts.
For the y-intercepts (when x = 0):
(0 + 1/2)^2 + (y - 3)^2 = 37
1/4 + (y - 3)^2 = 37
(y - 3)^2 = 148/4 - 1/4
(y - 3)^2 = 147/4
y - 3 = ±√(147/4)
y = 3 ± (√147)/2
Therefore, the y-intercepts are y = 3 + (√147)/2 and y = 3 - (√147)/2.