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Geometry Level 5

\[\begin{array} {} A: & x^2 + y^2 -x+6y+5 =0 \\ B: & x^2+y^2 +5x-8y+15 =0 \\ C: & x^2+y^2 +3x+2y+1= 0 \end{array} \]

Let $(h,k)$ be the center of the circle cutting the above three circles orthogonally, find the value of $h+k$ .

2 solutions

Ahmad Saad
Mar 29, 2017

Harsh Shrivastava
Mar 15, 2017

Hint: Prove that the required point is the radical center of the system.

Exactly The radical axis of any 2 circles passes through the centre of the required Circe so $A-B=0$ and $B-C=0$ .Solving $h,k$ = $(3,2)$

Spandan Senapati - 4 years, 2 months ago

Yep its highly over rated in my regard . its about solving two linear equations

3x-7y+5 = 0 and x-5y+7 = 0 :P

Prakhar Bindal - 4 years, 2 months ago

Ya but not all might know radical axis and may proceed along the usual lengthy way.

Spandan Senapati - 4 years, 2 months ago

I guess only jee ppl would know this property, others might do the lengthy way :P

Harsh Shrivastava - 4 years, 2 months ago

@Harsh Shrivastava – Yeah right! :P

Prakhar Bindal - 4 years, 2 months ago

Yes just we have to find the radical center.

Harry Jones - 3 years, 9 months ago

Don't think it's a level 5 problem thoug:P

Please Please - 3 years, 2 months ago