1. What is the equation of the circle with center (-4,-3) that passes through the point (6,2)?
A. (x-4)^2+(y-3)^2=25
B. (x-4)^2+(y-3)^2=125
C. (x+4)^2+(y+3)^2=25***
D. (x+4)^2+(y+3)^2=125
I think the answer is C but I'm not sure. Can someone help tell me if I'm correct or not.
The standard form equation of a circle:
( x - a ) ^ 2 + ( y - b ) ^ 2 = r ^ 2
a and b are the x and y coordinates of the center of the circle.
r = the radius of the circle.
In this case:
a = - 4 , b = - 3
[ x - ( - 4 ) ] ^ 2 + [ y - ( - 3 ) ] ^ 2 = r ^ 2
( x + 4 ) ^ 2 + ( y + 3 ) ^ 2 = r ^ 2
For x = 6 , y = 2
( x + 4 ) ^ 2 + ( y + 3 ) ^ 2 = r ^ 2
( 6 + 4 ) ^ 2 + ( 2 + 3 ) ^ 2 = r ^ 2
10 ^ 2 + 5 ^ 2 = r ^ 2
100 + 25 = r ^ 2
125 = r ^ 2
r ^ 2 = 125
The equation of the circle:
( x + 4 ) ^ 2 + ( y + 3 ) ^ 2 = 125
Answer D.
To determine the equation of a circle, you can start by using the general equation for a circle:
(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 given as (-4, -3), and it passes through the point (6, 2).
To find the radius, you need to find the distance between the center of the circle (-4, -3) and the given point (6, 2). You can use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
By substituting the values into the formula:
Distance = sqrt((6 - (-4))^2 + (2 - (-3))^2)
= sqrt((6 + 4)^2 + (2 + 3)^2)
= sqrt(10^2 + 5^2)
= sqrt(100 + 25)
= sqrt(125)
So the distance between the center (-4, -3) and the point (6, 2) is sqrt(125).
Since the radius is equal to the distance, we can conclude that the radius of the circle is sqrt(125).
Now, you can substitute these values into the general equation of a circle:
(x - (-4))^2 + (y - (-3))^2 = (sqrt(125))^2
(x + 4)^2 + (y + 3)^2 = 125
Therefore, the correct answer is option C: (x + 4)^2 + (y + 3)^2 = 125.