Question

show that the point (11,2) is the centre of the circle passing through the points (1,2),(3,-4) and (5,-6)
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Answer 1

Step-by-step explanation:

To prove that centre of the given cirle is (11,2)

Proof:

let the given points on the circle are A(1,2), B(3,-4) , C(5,-6)

let O(a,b) &r be the centre and radius of the cirle resply.,

=> the radii from O to A,B,C are equal

ie., OA = OB = OC = r

=>  (a-1)² + (b-2)²  = (a-3)² + (b-(-4))² = (a-5)² + (b-(-6))²

=> (a-1)² + (b-2)²  = (a-3)² + (b+4)² = (a-5)² + (b+6)²  ------- (1)

take 1st and 2nd

=> (a-1)² + (b-2)²  = (a-3)² + (b+4)²

=> a²+1-2a + b²+4-4b = a²+9-6a + b²+16+8b

=> 5-2a-4b = 25-6a+8b

=> -2a+6a -4b-8b= 25-5

=> 4a-12b = 20  (or) a-3b = 5   ------(2)

take 1st and 3rd

=> (a-1)² + (b-2)²  = (a-5)² + (b+6)²         => (11-1)²+(2-2)² = (11-5)²+(2+6)²

=> a²+1-2a + b²+4-4b = a²+25-10a + b²+36+12b

=> -2a+10a -4b-12b= 61-5

=> 8a-16b = 56  (or) 2a-4b =14   ------(3)

2*(2) - (3)

=> 2a-6b- 2a +4b = 10-14      

=> -2b = -4

=> b = 2

substitute the value of b in (2)  , a-3b = 5

=> a - 3(2) = 5

=> a-6 = 5

=> a = 5+6

=> a = 11

so the centre O (a,b) is O(11,2)

Hence proved

Alternate method:

Find the distances OA , OB and OC (radii) with O(11,2), A(1,2), B(3,-4) , C(5,-6)

using the distance formula.

if OA = OB = OC

then we can say that O is the centre of the Circle