Find the length of the line from A to C, which would be the radius using the distance formula:
Distance = √(-5 +1)^2 + 1+2)^2)
Distance = √(4^2 + 3^2)
Distance = √25
Distance = 5
The equation of a circle is written as (x-h)^2 + (y-k)^2 = r^2, where h and k is the center point X and y values and r is the radius.
The center of the circle is located at A ( -1,-2) and the radius is 5.
Replace h, k and r with those values:
(x - (-1))^2 + (y- (-2))^2 = 5^2
Simplify to get:
(x+1)^2 + (y+2)^2 = 25
The correct option for the equation of a circle A with radius segment AC is (x + 1)² + (y + 2)² = 25
The reason for the above selection is given as follows:
The given vertices of the triangle ΔABC, are:
A(-1, -2), B(-1, 1), and C(-5, 1)
The required parameter:
The equation of a circle with center at A and radius of segment AC
Method:
The length of segment AC which is the radius of the circle is found and
with the coordinates of point A and the calculated length of the radius, the
values are plugged into the general equation of a circle to arrive at the
correct answer option
The length of segment AC = √((-5 - (-1))² + (1 - (-2))²) = 5
∴ The length of segment AC = 5 = The length of the radius of circle A, r
r = 5
The general equation of a circle is (x - h)² + (y - k)² = r²
Where;
(h, k) = The coordinates of the center of the circle = Point A(-1, -2)
∴ h = -1, and k = -2
Plugging in the values of r, h, and k, in the equation of a circle gives;
(x - h)² + (y - k)² = r² where, h = -1, k = -2, and r = 5 gives;
(x - (-1))² + (y - (-2))² = 5²
We get;
(x + 1)² + (y + 2)² = 5² = 25
Therefore, the correct option is (x + 1)² + (y + 2)² = 25
Learn more about the equation of a circle here:
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