Basic Circle

Algebra Level 1

The radius of this circle can be expressed in the format A B A\sqrt { B } in simplest radical form

x 2 + y 2 − 4 x + 6 y − 11 = 0 { x }^{ 2 }+{ y }^{ 2 }-4x+6y-11=0

Evaluate A + B A + B


The answer is 8.

Dominick Hing by Dominick Hing , 23, USA

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1 solution

Dominick Hing
Oct 3, 2014

We need to complete the square twice. But first, it may help to rearrange the terms so that the x terms are together and the y terms are together:

x 2 − 4 x + y 2 + 6 y − 11 = 0 { x }^{ 2 }-4x+{ y }^{ 2 }+6y-11=0

Complete the Square (twice):

( x 2 − 4 x + 4 ) + ( y 2 + 6 y + 9 ) = 11 + 4 + 9 ({ x }^{ 2 }-4x+4)+{ (y }^{ 2 }+6y+9)=11+4+9

We added a 4 and a 9 to both sides to make two sets of perfect square trinomials on the left hand side

( x − 2 ) 2 + ( y + 3 ) 2 = 24 { ({ x }-2) }^{ 2 }+{ (y+3) }^{ 2 }=24

The radius is 24 = 2 6 \sqrt { 24 } =2\sqrt { 6 }

2 + 6 2 + 6 equals 8 8

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