Find the centre and radius of a circle 36x^2+24x+36y^2-36y-23=0
No need to solve for r. Follow the hint and complete the squares.
36x^2+24x+36y^2-36y-23=0
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 + 36/9 + 36/4
36(x + 1/3)^2 + 36(y - 1/2)^2 = 36
(x + 1/3)^2 + (y - 1/2)^2 = 1
To find the center and radius use the formula for the equation of a circle (x-h)^2 + (y-k)^2=r^2 where (h,k) is the center. and then solve for r by plugging in a point for x and y with using the center for h,k
btw find the equation by factoring and completing the square if necessary
Ah, the wonderful world of circles! Let's get started, shall we?
To find the center and radius of a circle given its equation, we need to put it in the standard form: (x - h)^2 + (y - k)^2 = r^2.
So, let's complete the square for both the x and y terms:
For the x terms:
36x^2 + 24x = 36(x^2 + (2/3)x) = 36(x^2 + (2/3)x + 1/9 - 1/9)
And for the y terms:
36y^2 - 36y = 36(y^2 - y) = 36(y^2 - y + 1/4 - 1/4)
Now, let's write the equation in standard form:
36(x^2 + (2/3)x + 1/9) + 36(y^2 - y + 1/4) = 23 + 36 * (1/9) + 36 * (1/4)
To simplify a bit:
36(x^2 + (2/3)x + 1/9) + 36(y^2 - y + 1/4) = 23 + 4 + 9
Combining like terms:
36(x^2 + (2/3)x + 1/9) + 36(y^2 - y + 1/4) = 36
Dividing by 36 on both sides:
(x^2 + (2/3)x + 1/9) + (y^2 - y + 1/4) = 1
Now, we can rewrite this in the standard form:
(x + 1/3)^2 + (y - 1/2)^2 = 1
So, the center of the circle is (-1/3, 1/2), and the radius is the square root of 1, which is just 1.
Voila! Circle deciphered!
To learn
To find the center and radius of the given circle, follow the steps below:
Step 1: Rewrite the equation in the standard form: (x - h)^2 + (y - k)^2 = r^2
Given equation: 36x^2 + 24x + 36y^2 - 36y - 23 = 0
Rearrange the equation by grouping the x-terms and y-terms separately and completing the square.
36x^2 + 24x + 36y^2 - 36y = 23
Divide throughout the equation by the coefficient of x^2 and y^2 to make the leading coefficients 1:
x^2 + (24/36)x + y^2 - (36/36)y = 23/36
Simplify:
x^2 + (2/3)x + y^2 - y = 23/36
Step 2: Now, focus on completing the square for x and y terms. To complete the square, follow these steps separately for the x and y terms:
For the x-term: Take half of the coefficient of x (which is 2/3), square it, and add it to both sides of the equation.
x^2 + (2/3)x + (2/3)^2 = 23/36 + (2/3)^2
x^2 + (2/3)x + 4/9 = 23/36 + 4/9
Simplify:
x^2 + (2/3)x + 4/9 = 29/36
For the y-term: Take half of the coefficient of y (which is -1), square it, and add it to both sides of the equation.
y^2 - y + (-1/2)^2 = 23/36 + (-1/2)^2
y^2 - y + 1/4 = 23/36 + 1/4
Simplify:
y^2 - y + 1/4 = 29/36
Step 3: Now, rewrite the equation using the completed square form for both x and y terms:
(x + 1/3)^2 + (y - 1/2)^2 = 29/36
Comparing this with the standard form (x - h)^2 + (y - k)^2 = r^2, we can determine that the center of the circle is (-1/3, 1/2) and the radius is sqrt(29/36).
So, the center of the circle is (-1/3, 1/2) and the radius is sqrt(29/36).