Find the coordinates of the centre of the circle passing through A(5,1), B(-3,-7) and C(7,-1) - Maths - Coordinate Geometry

Question

Find the coordinates of the centre of the circle passing through
A(5,1), B(-3,-7) and C(7,-1)

Ajanta Trivedi · Accepted Answer

Find the coordinates of the centre of the circle passing through A(5,1), B(-3,-7) and C(7,-1).

let the coordinates of the centre be O(h,k).

the distance from the centre to a point on the circumference is equal to radius.

therefore OA=OB=OC $\Rightarrow O A^{2} = O B^{2} = O C^{2}$

now from distance formula

$O A^{2} = O B^{2} (h - 5)^{2} + (k - 1)^{2} = (h + 3)^{2} + (k + 7)^{2} h^{2} + 25 - 10 h + k^{2} + 1 - 2 k = h^{2} + 9 + 6 h + k^{2} + 49 + 14 k 26 - 10 h - 2 k = 58 + 6 h + 14 k = 0 - 16 h - 16 k = 32 h + k = - 2............ (1)$

now taking $O B^{2} = O C^{2}$

$(h + 3)^{2} + (k + 7)^{2} = (h - 7)^{2} + (k + 1)^{2} h^{2} + 9 + 6 h + k^{2} + 49 + 14 k = h^{2} + 49 - 14 h + k^{2} + 1 + 2 k 58 + 6 h + 14 k = 50 - 14 h + 2 k 20 h + 12 k = - 8 5 h + 3 k = - 2......... (2)$

now solving equation (1) and (2),

multiply (1) by 3 and subtract from (2)

$5 h + 3 k = - 2 3 h + 3 k = - 6 2 h = 8 \Rightarrow h = \frac{4}{2} = 2$ substitute the value of h in (1)

$2 + k = - 2 k = - 2 - 2 = - 4$

hence the coordinates of the centre is (2,-4).

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