State the center and the radius of the circle represented by the equation
(x + 1)^2 + (y – 5)^2 = 64
in the general equation of a circle
(x-h)^2 + (y-k)^2 = r^2
the centre is (h,k) and the radius is r
can you match this with your given equation?
ok based on that I would say the radius is 8 cause 8*8 = 64 and the center is (1,5)? Or do the signs change because of the equation
(x-h)^2 + (y-k)^2 = r^2 making the center (-1,5)?
radius is correct,
the centre is (-1,5)
That is what I thought, so the standard form would be
(x-(-1]^2 + (y-5)^2 = 64
yes,
I just mentally change the sign I see in the bracket
Great! Thanks for the help! So R= 8 and the center is (-1,5) got it! Thanks again!
To determine the center and radius of the circle represented by the equation, we need to rewrite the equation in standard form, which is (x - h)^2 + (y - k)^2 = r^2, where the center of the circle is (h, k) and the radius is r.
Given equation: (x + 1)^2 + (y - 5)^2 = 64
To rewrite the equation in standard form, we expand the square terms:
(x + 1)(x + 1) + (y - 5)(y - 5) = 64
Simplifying further:
(x^2 + 2x + 1) + (y^2 - 10y + 25) = 64
Rearranging the terms:
x^2 + 2x + y^2 - 10y + 26 = 64
Moving the constant term from the right side to the left side:
x^2 + 2x + y^2 - 10y + 26 - 64 = 0
x^2 + 2x + y^2 - 10y - 38 = 0
Now, we complete the square separately for the x-term and y-term. For the x-term:
x^2 + 2x
To complete the square, we take half of the coefficient of x (which is 2), square it (2^2 = 4), and add it to the equation.
So, we add 4 to both sides:
x^2 + 2x + 4
For the y-term:
y^2 - 10y
To complete the square, we take half of the coefficient of y (which is -10), square it (-10/2)^2 = 25), and add it to the equation.
So, we add 25 to both sides:
y^2 - 10y + 25
Now, the equation becomes:
x^2 + 2x + 4 + y^2 - 10y + 25 - 38 = 0
Simplifying further:
x^2 + 2x + y^2 - 10y - 9 = 0
Rearranging the terms:
(x^2 + 2x) + (y^2 - 10y) - 9 = 0
Now, we can rewrite the equation in standard form:
(x^2 + 2x + 1) + (y^2 - 10y + 25) - 9 - 1 = 0
(x + 1)^2 + (y - 5)^2 - 9 - 1 = 0
(x + 1)^2 + (y - 5)^2 = 10
Comparing this equation with the standard form equation (x - h)^2 + (y - k)^2 = r^2, we can see that the center of the circle is (-1, 5) and the radius is the square root of 10.