Write the equation of the circle in standard form. Find the center, radius, intercepts, and graph the circle.
x^2+y^2-12x-4x+36=0
How on earth do I do this?
I meant to say *4y instead of 4x
x^2+y^2-12x-4y+36=0
you will have to complete the square and put the equation into standard form
x^2 - 12x + .... + y^2 - 4y + .... = -36 + .... + ....
x^2 - 12x + 36 + y^2 - 4y + 4 = -36 + 36 + 4
(x-6)^2 + (y-2)^2 = 4
so we have a circle with centre at (6,2) and radius 2
for x-intercept, let y = 0
(x-6)^2 + 4 = 4
x-6 = 0
x = 6
The circle touched the x-axis at (6,0)
for the y-intercept, let x = 0
36 + (y-2)^2 = 4
(y-2)^2 = -32
this has no solution, we can't take the √ of a
No y-intercepts
Your sketch will confirm that.
To write the equation of a circle in standard form, you need to complete the square for both the x and y variables. Here's how you can do it step by step:
Step 1: Rearrange the equation to group the x-terms and y-terms together:
x^2 - 12x + y^2 - 4y + 36 = 0
Step 2: Complete the square for the x-terms. To do this, take half of the coefficient of x (-12), square it ((-12)/2)^2 = 36), and add it to both sides of the equation:
x^2 - 12x + 36 + y^2 - 4y + 36 = 36 + 36
Simplifying further:
(x - 6)^2 + y^2 - 4y + 36 = 72
Step 3: Now complete the square for the y-terms. Take half of the coefficient of y (-4), square it ((-4)/2)^2 = 4), and add it to both sides of the equation:
(x - 6)^2 + y^2 - 4y + 4 = 72 + 4
Simplifying further:
(x - 6)^2 + (y - 2)^2 = 76
So, the equation of the circle in standard form is (x - 6)^2 + (y - 2)^2 = 76.
To find the center, we look at the values inside the parentheses of (x - h)^2 and (y - k)^2. The center of the circle is given by the coordinates (h, k). In this case, the center is (6, 2).
The radius of the circle is the square root of the constant term on the right side of the equation. So, the radius is √76, which is approximately 8.72.
To find the intercepts, we set either x or y to zero and solve for the other variable. Setting x = 0, we have:
(0 - 6)^2 + (y - 2)^2 = 76
36 + (y - 2)^2 = 76
(y - 2)^2 = 40
Taking the square root of both sides:
y - 2 = ±√40
y = 2 ± 2√10
So, the y-intercepts are y = 2 + 2√10 and y = 2 - 2√10.
To graph the circle, plot the center at (6, 2) and draw a circle with a radius of approximately 8.72 units around it.