write the standard form of the equation of the circle that passes through the given point (8,0) and whose center is the origin
The radius r is the distance between (8,0) and the origin (0,0).
After that, use the standard form given in the response to:
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To find the standard form of the equation of the circle that passes through the point (8,0) and has its center at the origin, we 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 its radius.
Since the center of the circle is the origin (0,0), the equation simplifies to:
x^2 + y^2 = r^2
Now, we need to find the value of r, which is the distance from the center to the given point (8,0). We can use the distance formula to calculate this:
r = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values, we have:
r = √((8 - 0)^2 + (0 - 0)^2)
r = √(8^2 + 0^2)
r = √(64)
r = 8
Therefore, the equation of the circle in standard form is:
x^2 + y^2 = 8^2
x^2 + y^2 = 64
To find the standard form of the equation of a circle, we need the coordinates of the center and the radius. In this case, we are given that the center of the circle is the origin, which has coordinates (0, 0).
We also know that the circle passes through the point (8, 0).
The distance from the center of the circle to any point on the circle is equal to the radius. Therefore, the radius of the circle is the distance between the origin (0, 0) and the point (8, 0). We can calculate the distance using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, x1 = 0, y1 = 0 (origin), x2 = 8, and y2 = 0. Plugging these values into the distance formula:
Distance = sqrt((8 - 0)^2 + (0 - 0)^2)
Distance = sqrt(8^2 + 0^2)
Distance = sqrt(64 + 0)
Distance = sqrt(64)
Distance = 8
Therefore, the radius of the circle is 8.
Now, we can use the standard form of the equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the coordinates of the center, and r represents the radius.
In this case, the center is (0, 0) and the radius is 8. Plugging these values into the equation:
(x - 0)^2 + (y - 0)^2 = 8^2
x^2 + y^2 = 64
Therefore, the standard form of the equation of the circle that passes through the point (8, 0) and whose center is the origin is x^2 + y^2 = 64.