Find the centre of the circle 36x^2+36y^2-24x-36y-23=0
complete the squares and it becomes clear:
36x^2+36y^2-24x-36y-23=0
36x^2-24x + 36y^2-36y = 23
36(x^2 - 2/3 x) + 36(y^2-y) = 23
36(x^2 - 2/3 x + 1/9) = 36(y^2 - y + 1/4) = 23 + 36(1/9) + 36(1/4)
36(x - 1/3)^2 + 36(y - 1/2)^2 = 36
(x - 1/3)^2 + (y - 1/2)^2 = 1
thanks alot
To find the center of the circle, we need to rewrite the equation of the circle in standard form, which is (x-h)^2 + (y-k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius.
Let's rewrite the equation step-by-step:
1. Group the x-terms and y-terms separately:
36x^2 - 24x + 36y^2 - 36y - 23 = 0
2. Complete the square for the x-terms:
To complete the square for the x-terms, take half of the coefficient of x (-24/2 = -12), square it, and add it to both sides of the equation.
36x^2 - 24x + (-12)^2 + 36y^2 - 36y - 23 = (-12)^2
This simplifies to:
36x^2 - 24x + 144 + 36y^2 - 36y - 23 = 144
3. Repeat the same process for the y-terms:
36y^2 - 36y + (-18)^2 = 0
This simplifies to:
36y^2 - 36y + 324 + 144 + 144 = 144
4. Combine like terms:
36(x^2 - (2/3)x) + 36(y^2 - y) + 245 = 0
5. Factor out the common coefficients:
36(x^2 - (2/3)x) + 36(y^2 - y) + 245 = 0
6. Complete the square for each binomial:
(x - 1/3)^2 - 1/9 + (y - 1/2)^2 - 1/4 + 245 = 0
This further simplifies to:
(x - 1/3)^2 + (y - 1/2)^2 + 245 - 1/9 - 1/4 = 0
7. Combine the constants:
(x - 1/3)^2 + (y - 1/2)^2 + 2207/36 = 0
Now, we can see that the equation is in the form (x - h)^2 + (y - k)^2 = r^2. Comparing it to the standard form, we can determine the values of (h, k) as the center of the circle.
From this equation, we can conclude that the center of the circle is at (-1/3, 1/2).
Well, finding the center of a circle is no circus act! We just need to remember that a circle's equation is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. So, let's juggle with your equation a bit!
Starting with 36x^2 + 36y^2 - 24x - 36y - 23 = 0, let's group the x and y terms:
(36x^2 - 24x) + (36y^2 - 36y) = 23
Now, we can complete the square by adding and subtracting the appropriate terms:
(36x^2 - 24x + 36) + (36y^2 - 36y + 81) = 23 + 36 + 81
Simplifying, we get:
36(x^2 - (2/3)x + 1) + 36(y^2 - y + 9/4) = 140
Dividing by 36:
x^2 - (2/3)x + 1 + y^2 - y + 9/4 = 140/36
Combining like terms:
x^2 - (2/3)x + y^2 - y + 25/36 = 35/9
Now, we can complete the square for both x and y:
(x - 1/3)^2 + (y - 1/2)^2 = 35/9 - 25/36
Simplifying, we get:
(x - 1/3)^2 + (y - 1/2)^2 = 35/36
Voila! We've found the center! The center of the circle is (1/3, 1/2). So, there you have it, clowning around and solving math problems at the same time!
To find the center of a circle given its equation, we need to rewrite the equation in the standard form of a circle, which is (x - h)^2 + (y - k)^2 = r^2. In this form, the center of the circle is at point (h, k), and the radius of the circle is r.
Let's rearrange the equation provided to match the standard form:
36x^2 + 36y^2 - 24x - 36y - 23 = 0
First, group the x-terms and the y-terms:
(36x^2 - 24x) + (36y^2 - 36y) - 23 = 0
Next, complete the square separately for the x and y terms.
For the x-terms:
Take half of the coefficient of x (-24/2 = -12), square it (-12)^2 = 144, and add it inside the parentheses. Since we added it inside the parentheses, we also need to subtract it outside the parentheses to maintain balance:
(36x^2 - 24x + 144) + (36y^2 - 36y) - 23 - 144 = 0
For the y-terms:
Take half of the coefficient of y (-36/2 = -18), square it (-18)^2 = 324, and add it inside the parentheses:
(36x^2 - 24x + 144) + (36y^2 - 36y + 324) - 23 - 144 = 0
Now, we can rewrite the expression as perfect squares:
(6x - 12)^2 + (6y - 18)^2 - 23 - 144 = 0
Simplify further:
(6x - 12)^2 + (6y - 18)^2 = 167
Now that we have the equation in standard form, we can identify the center and the radius.
The center of the circle is given by the point (h, k), which is (12/6, 18/6).
Simplifying further, we get the center as (2, 3).
Thus, the center of the circle is at (2, 3).