find the coordinates of the centre of the circle passing through the point (0,0),(-2,1)and(-3,2).also find the radius
Answers
:-)
Given,
The coordinates of the points on the circumference of the circle are = (0,0) , (-2,1) and (-3,2)
To find,
The coordinates of the centre of the circle and the length of the radius of the given circle.
Solution,
We can simply solve this mathematical problem by using the following mathematical process.
Let,
Point A = (0,0)
Point B = (-2,1)
Point C = (-3,2)
Point O = (x,y) = centre of the circle
As the radius length is always constant,
Thus,
OA = OB = OC
So,
OA² = OB²
By applying the distance formula, we get,
(x-0)²+(y-0)² = (x+2)²+(y-1)²
x²+y² = x²+4x+4+y²-2y+1
x²-x²+y²-y²-4x+2y = 4+1
2y-4x = 5 ....(1)
Similarly,
OB² = OC²
(x+2)²+(y-1)² = (x+3)²+(y-2)²
x²+4x+4+y²-2y+1 = x²+6x+9+y²-4y+4
x²+4x+y²-2y-x²-6x-y²+4y = 9+4-4-1
2y-2x = 8 ....(2)
Subtracting (2) from (1), we get,
2y-4x = 5
-2y+2x = -8
___________
-2x = -3
x = 3/2
Putting the value of x in (1), we get,
2y - (4×3/2) = 5
2y-6 = 5
2y = 11
y = 11/2
Coordinate of centre = (3/2, 11/2)
Radius (OA) = √(3/2-0)²+ (11/2-0)² = √(9/4 + 121/4) = √130/4 = √32.5 = 5.7 units (approx)
Hence, the coordinate of the centre of the circle is (3/2,11/2) and the length of radius is approximately 5.7 units.