what is the equation of the circle with center (-4,-3) that passes through the point (6,2)
The equation of a circle with center (a,b) and radius r is:
(x-a)^2 + (y-b)^2 = r^2
We are given that the center of the circle is (-4,-3) and it passes through the point (6,2). Using the formula above, we can plug in these values to find the equation of the circle:
(x-(-4))^2 + (y-(-3))^2 = r^2
Simplifying this, we get:
(x+4)^2 + (y+3)^2 = r^2
We still need to find the value of r. Since we know that the circle passes through the point (6,2), we can substitute these values in for x and y:
(6+4)^2 + (2+3)^2 = r^2
Simplifying this, we get:
100 + 25 = r^2
125 = r^2
Taking the square root of both sides, we get:
r = √125 = 5√5
Now we can plug in the value of r and simplify the equation of the circle:
(x+4)^2 + (y+3)^2 = (5√5)^2
(x+4)^2 + (y+3)^2 = 125
So the equation of the circle with center (-4,-3) that passes through the point (6,2) is:
(x+4)^2 + (y+3)^2 = 125
To find the equation of a circle, we can use the formula:
(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 of the circle is (-4, -3) and the circle passes through the point (6, 2).
Step 1: Find the radius of the circle
The radius can be found by using the distance formula between the center and a point on the circle. In this case, we can use the distance formula:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we have:
r = sqrt((6 - (-4))^2 + (2 - (-3))^2)
r = sqrt((10)^2 + (5)^2)
r = sqrt(100 + 25)
r = sqrt(125)
r = 5 * sqrt(5)
Step 2: Substitute the values into the equation
Now that we have the center (-4, -3) and the radius 5 * sqrt(5), we can substitute these values into the equation:
(x - (-4))^2 + (y - (-3))^2 = (5 * sqrt(5))^2
Simplifying:
(x + 4)^2 + (y + 3)^2 = 125
So, the equation of the circle with center (-4, -3) that passes through the point (6, 2) is (x + 4)^2 + (y + 3)^2 = 125.