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
(a)
to write this in standard form, you use completing the square for both variables x and y.
x^2 + y^2 + 8y - 65 = 0
x^2 + (y^2 + 8y + 16) - 65 - 16 = 0
x^2 + (y + 4)^2 - 81 = 0
x^2 + (y + 4)^2 = 81
x^2 + (y + 4)^2 = 9^2
(b)
Center: C(0, -4)
Radius = 9
to get intercepts,
x-intercept: let y=0 and solve for x
y-intercept: let x=0 and solve for y
(c)
with center and radius known, you can graph this on a graphing paper. use a compass to draw the circle.
hope this helps~ :)
(a) To put the equation in standard form, we need to complete the square for the y-terms. So, let's begin by rearranging the equation:
x^2 + y^2 + 8y - 65 = 0
Rearranging terms, we have:
x^2 + (y^2 + 8y) - 65 = 0
Now, let's complete the square for the y-terms.
We add and subtract the square of half the coefficient of y (which is 4) inside the parentheses:
x^2 + (y^2 + 8y + 16 - 16) - 65 = 0
Simplifying further:
x^2 + (y^2 + 8y + 16) - 16 - 65 = 0
x^2 + (y + 4)^2 - 81 = 0
So, the equation in standard form is:
x^2 + (y + 4)^2 = 81
(b) From the equation, we can determine the center, radius, and intercepts.
The center is at the coordinates (h, k), which in this case are (0, -4).
The radius is √(r^2) = √81 = 9.
To find the intercepts, we set x or y to zero and solve for the other variable:
Setting x = 0, we have:
0^2 + (y + 4)^2 = 81
(y + 4)^2 = 81
Taking the square root of both sides:
y + 4 = ±9
y = -4 ± 9
So, the y-intercepts are y = -4 + 9 = 5 and y = -4 - 9 = -13.
Setting y = 0, we have:
x^2 + (-4)^2 = 81
x^2 + 16 = 81
x^2 = 81 - 16
x^2 = 65
Taking the square root of both sides:
x = ±√65
So, the x-intercepts are x = √65 and x = -√65.
Therefore, the center is at (0, -4), the radius is 9, and the intercepts are (0, 5), (0, -13), (√65, 0), and (-√65, 0).
(c) Unfortunately, as a text-based AI, I am unable to submit a graph. However, you can graph the equation by plotting the center at (0, -4), drawing a circle with a radius of 9 centered at the origin, and marking the intercepts.
To write the equation in standard form, you need to complete the square for the variables x and y.
(a) Equation in standard form:
To complete the square for x^2, we need to add (8/2)^2 = 16 to both sides of the equation:
x^2 + y^2 + 8y - 65 + 16 = 16
x^2 + y^2 + 8y - 49 = 16
To complete the square for y^2 + 8y, we need to add (8/2)^2 = 16 to both sides of the equation:
x^2 + y^2 + 8y - 49 + 16 = 16 + 16
x^2 + y^2 + 8y - 33 = 32
Simplifying the equation:
x^2 + y^2 + 8y = 65
(b) Center, Radius, and Intercepts:
To determine the center and the radius of the circle represented by the equation, we need to rewrite the equation in the general form of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle.
Let's group the x and y terms:
x^2 + y^2 + 8y = 65
Now, we will complete the square for the y terms:
x^2 + (y^2 + 8y + 16) = 65 + 16
x^2 + (y + 4)^2 = 81
Comparing it with the general form of a circle:
(x - 0)^2 + (y + 4)^2 = 9^2
The center is at (0, -4), and the radius is 9.
To find the x and y intercepts, we set the respective variable to 0.
- Setting y = 0:
x^2 + 0 + 0 = 65
x^2 = 65
x = ±√65
So, the x-intercepts are (√65, 0) and (-√65, 0).
- Setting x = 0:
0 + y^2 + 8y = 65
y^2 + 8y - 65 = 0
To solve for y, we can either factor or use the quadratic formula. Assuming factoring is not possible, we use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Using a = 1, b = 8, and c = -65:
y = (-8 ± √(8^2 - 4(1)(-65))) / (2(1))
y = (-8 ± √(64 + 260)) / 2
y = (-8 ± √324) / 2
y = (-8 ± 18) / 2
The y-intercepts are (-13, 0) and (5, 0).
(c) Graph:
As a text-based AI, I cannot display the graph directly here. However, you can plot the graph on a graphing software or use online graphing tools. Simply substitute various values of x into the equation, solve for y, and plot the points. Connect the plotted points to form a circle with the given center and radius.