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Question

Find the coordinates of the centre of a circle which passes through the points A(1,2), B(3,-4) and C(5,-6). 


  1. (11,2)

  2. (11,2)

  3. (11,2)

  4. (11,2)


Solution

The correct option is A

(11,2)


Let the coordinates of the centre of the circle be O(a,b)

Since the circle passes through the 3 points  A(1,2). B(3,-4) and C(5,-6), their distance from the centre will be equal to the radius.

Therefore OA2=OB2=OC2

Distance Formula = (x2x1)2+(y2y1)2

OA2 = (a1)2+(b2)2 ............(i)

OB2 = (a3)2+(b+4)2...........(ii)

OC2 = (a5)2+(b+6)2...............(iii)

Equating (i) and (ii) we get

 (a1)2+(b2)2
=  (a3)2+(b+4)2

  a2+12a+b2+44b
=  a2+96a+b2+16+8b

  2a+54b =  6a+25+8b

  4a12b =  20  ..........(iv)

Equating (ii) and (iii) we get

(a3)2+(b+4)2
(a5)2+(b+6)2

  a2+96a+b2+16+8b
= a2+2510a+b2+36+12b

  4a4b=36 ........(v)

Solving (iv) and (v) we get a=11,b=2

Therefore, centre of the circle is (11,2).

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