Can anyone explain to me how to graph a circle if it has a square root radius?
Example: x^2+y^2-7x=12-4y
The radius is √113/2, but I don't know how to graph it.
x^2 - 7 x = 12 - 4 y -y^2
x^2 -7x + 49/4 = 97/4 -4y -y^2
(x-7/2)^2 + y^2 + 4y =97/4
(x-7/2)^2 + y^2 + 4y + 4 =97/4+16/4
(x-7/2)^2 + (y+2)^2 = 113/4 sure enough
center at x=3.5 , y = -2
r = (1/2)sqrt(113) = 5.32
x^2+y^2-7x=12-4y
x^2 - 7x + 49/4 + y^2 + 4y + 4 = 12 + 49/4 + 4
(x - 7/2)^2 + (y+2)^2 = 113/4 = √113/2
you had that
so the centre is (7/2 , -2) and the radius is √113/2
which is appr 5.315
set you compass at appr 5.3 units, and with the above centre, draw your circle
To graph a circle, you need to determine the center and radius of the circle from the given equation.
In the equation given, x^2 + y^2 - 7x = 12 - 4y, you can rearrange it into the standard form of a circle equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is its radius.
In order to do this, complete the square for both the x and y terms separately.
For the x-terms:
1. Move the constant term (12) to the right side of the equation: x^2 + y^2 - 7x + 4y = 12.
2. Group the x-terms and complete the square:
- Rearrange the x-terms: x^2 - 7x + y^2 + 4y = 12.
- Take half of the x-coefficient (-7/2) and square it: (-7/2)^2 = 49/4.
- Add this value to both sides of the equation: x^2 - 7x + 49/4 + y^2 + 4y = 12 + 49/4.
3. Rewrite the equation: (x^2 - 7x + 49/4) + (y^2 + 4y) = 12 + 49/4.
4. Simplify the equation: (x - 7/2)^2 + (y^2 + 4y) = 113/4.
Now, you can see that the equation is in the standard form, where the x-coordinate of the center is 7/2 and the y-coordinate is 0. The radius, r, can be found by taking the square root of the constant term obtained (113/4) and dividing it by 2.
So, the center of the circle is (7/2, 0), and the radius is √(113/4)/2 = √113/4.
To graph the circle:
1. Plot the center point, which is (7/2, 0), on a coordinate plane.
2. Use the radius to determine the distance from the center to points on the circle. In this case, the radius is √113/4.
3. Starting from the center, mark two points on the circle that are √(113/4)/2 units to the left and right of the center. Additionally, mark two points that are √(113/4)/2 units above and below the center.
4. Connect these four points using a smooth curve. This curve represents the graph of the circle with a square root radius.
Thus, you have graphed the given circle with the equation x^2 + y^2 - 7x = 12 - 4y.