Write the equation of a circle in standard Find the center, radius intercepts and graph the circle. .........x^2+y^2-12x-4y+36=0
x^2 -12 x = -y^2 + 4 y -36
x^2 -12 y + 36 = -y^2 + 4 y
-(x-6)^2 = y^2-4y
-(x-6)^2 + 4 = y^2 - 4 y + 4
-(x-6)^2 + 4 = (y-2)^2
(x-6)^2 + (y-2)^2 = 4
center at (6,2) radius 2
you do the rest
Find the center (h,k) and the radius r of the circle
5x2−8x+5y2−5y−6=0.
To find the equation of a circle in standard form, we have to complete the square for both x and y variables.
Given: x^2 + y^2 - 12x - 4y + 36 = 0
First, let's group the x terms and the y terms separately:
(x^2 - 12x) + (y^2 - 4y) + 36 = 0
Now, to complete the square for the x terms, we take the coefficient of x (-12), divide it by 2 (-6), square it (-6^2 = 36), and add it to both sides:
(x^2 - 12x + 36) + (y^2 - 4y) + 36 = 36 + 36
(x - 6)^2 + (y^2 - 4y) + 36 = 72
Next, let's complete the square for the y terms. We take the coefficient of y (-4), divide it by 2 (-2), square it (-2^2 = 4), and add it to both sides:
(x - 6)^2 + (y^2 - 4y + 4) + 36 = 72 + 4
(x - 6)^2 + (y - 2)^2 + 36 = 76
Now, we can rewrite the equation in standard form:
(x - 6)^2 + (y - 2)^2 = 76 - 36
(x - 6)^2 + (y - 2)^2 = 40
So, the equation of the circle in standard form is (x - 6)^2 + (y - 2)^2 = 40.
To find the center and radius of the circle, we can identify them from the standard form equation.
The center of the circle is given by (h, k), where h and k are the x and y coordinates of the center. In this case, the center is (6, 2).
The radius of the circle is the square root of the value on the right side of the equation. So, the radius of this circle is √40.
To find the x and y intercepts of the circle, we can set x or y equal to zero and solve for the other variable.
Setting x = 0, we have:
(0 - 6)^2 + (y - 2)^2 = 40
36 + (y - 2)^2 = 40
(y - 2)^2 = 4
Taking the square root, we get:
y - 2 = ±2
y = 2 ± 2
So, the y-intercepts are y = 4 and y = 0.
Setting y = 0, we have:
(x - 6)^2 + (0 - 2)^2 = 40
(x - 6)^2 + 4 = 40
(x - 6)^2 = 36
Taking the square root, we get:
x - 6 = ±6
x = 6 ± 6
So, the x-intercepts are x = 12 and x = 0.
Now, to graph the circle, we can plot the center (6, 2), and then measure the radius √40 (approximately 6.32) in each direction to plot points on the circle. Finally, we can draw a smooth curve through these points to form the circle.