Determine the equation of a circle given the center (0,0) and a point on the circumference (5, 3)
the general equation for a circle of radius r, centered at (h,k)
... (x - h)^2 + (y - k)^2 = r^2
x^2 + y^2 = 5^2 + 3^3
To determine the equation of a circle given the center and a point on the circumference, you can use the formula for the equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the center of the circle and r represents the radius of the circle.
In this case, the center of the circle is (0, 0), and a point on the circumference is (5, 3).
Using the formula, we can substitute the values:
(x - 0)^2 + (y - 0)^2 = r^2
Simplifying, we get:
x^2 + y^2 = r^2
Next, we need to find the value of r, the radius. We can use the distance formula to calculate the distance between the center point (0, 0) and the point on the circumference (5, 3).
The distance formula is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the values from the given points, we have:
d = sqrt((5 - 0)^2 + (3 - 0)^2)
= sqrt(25 + 9)
= sqrt(34)
Since the radius is equal to the distance from the center to a point on the circumference, we have:
r = sqrt(34)
Therefore, the equation of the circle is:
x^2 + y^2 = 34
To determine the equation of a circle, we need to use the general equation of a circle, which is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle, and r represents the radius. In this case, the center is given as (0, 0), and we are given a point on the circumference as (5, 3).
Step 1: Find the radius (r)
The radius can be found by calculating the distance from the center (0, 0) to the given point (5, 3) using the distance formula:
r = √((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we have:
r = √((5 - 0)^2 + (3 - 0)^2)
r = √(25 + 9)
r = √34
Step 2: Substitute the center and radius values into the general equation
Now that we have the center (h, k) = (0, 0), and the radius r = √34, we can substitute these values into the equation:
(x - 0)^2 + (y - 0)^2 = (√34)^2
x^2 + y^2 = 34
Therefore, the equation of the circle with the center (0,0) and a point on the circumference (5, 3) is x^2 + y^2 = 34.