Given the equation x^2 +y^2+8y-65=0
(a) Write the equation in standard form.
(b) State the center, radius, and intercepts.
(c) Submit the graph
rearrange terms to complete the squares:
x^2 + (y^2 + 8y + 16) = 65 + 16
x^2 + (y+4)^2 = 9^2
center: (0,-4)
radius: 9
y-intercepts: (0,-13) and (0,5)
x-intercepts: (-√65,0) and (√65,0)
(a) To write the equation x^2 +y^2+8y-65=0 in standard form, we need to complete the square for the y terms.
Start by rearranging the equation:
x^2 + (y^2 + 8y) = 65
Now, we want to complete the square for the y terms. Take half of the coefficient of y (which is 8) and squared:
8/2 = 4
4^2 = 16
Add and subtract 16 inside the parentheses:
x^2 + (y^2 + 8y + 16 - 16) = 65
Group the terms inside the parentheses:
x^2 + (y^2 + 8y + 16) - 16 = 65
Rearrange to isolate the perfect square terms:
x^2 + (y^2 + 8y + 16) = 65 + 16
Combine the constants on the right side of the equation:
x^2 + (y^2 + 8y + 16) = 81
Factor the perfect square inside the parentheses:
x^2 + (y + 4)^2 = 9^2
Now, the equation is in standard form:
x^2 + (y + 4)^2 = 81
(b) From the standard form equation, we can identify the values for the center, radius, and intercepts.
Center: In the standard form equation, the center (h, k) of the circle is given by the opposite of the x-coefficient and the opposite of the y-coefficient of the perfect square term. In this case, the center (h, k) is (0, -4).
Radius: The radius of the circle is the square root of the constant term on the right side of the equation. In this case, the radius is √81, which simplifies to 9.
Intercepts: To find the x-intercepts, set y = 0 in the original equation and solve for x. To find the y-intercepts, set x = 0 and solve for y. In this case, we substitute y = 0 and x = 0 into the equation:
For x-intercepts:
0^2 + y^2 + 8y - 65 = 0
y^2 + 8y - 65 = 0
Solve this quadratic equation to find the x-intercepts.
For y-intercepts:
x^2 + 0^2 + 8(0) - 65 = 0
x^2 - 65 = 0
Solve this quadratic equation to find the y-intercepts.
(c) Unfortunately, as a text-based AI, I cannot directly submit a graph. However, you can input the equation x^2 + (y + 4)^2 = 81 into graphing software or a graphing calculator to visualize the circle described by the equation. The center is located at (0, -4), and the radius is 9 units.