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Find the angle between the circles x^{2}+y^{2}+2x+4y+1=0\, and x^{2}+y^{2}+4x+2y+4=0\,

For the first circle centre and radius are

A(-1,-2),r_{1}={\sqrt  {1+4-1}}=2\,

Similarly for the second circle

B(-2,-1),r_{2}={\sqrt  {4+1-4}}=1\,

Distance between the centres is

d=AB={\sqrt  {(-1+2)^{2}+(-2+1)^{2}}}={\sqrt  {2}}\,

If theta is the angle between the circles,

\cos \theta ={\frac  {d^{2}-r_{1}^{{2}}-r_{2}^{{2}}}{2r_{1}r_{2}}}\,

\cos \theta ={\frac  {2-4-1}{2\cdot 2\cdot 1}}={\frac  {-3}{4}}\,

\theta =\arccos({\frac  {-3}{4}})\,

Main Page:Geometry:Circles