Geo5.1.20

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Show that the equation of common tangents to the circle x^{2}+y^{2}=2a^{2}\, and the parabola y^{2}=8ax\, is y=\pm (x+2a)\,

Given the equation of the parabola is y^{2}=8ax\, and the circle is x^{2}+y^{2}=2a^{2}\,

Any tangent of the parabola is y=mx+2alm,mx-y+2alm=0\,

This is also a tangent to the circle,

{\frac  {{\frac  |{2a}}{m}|}{{\sqrt  {1+m^{2}}}}}=a{\sqrt  {2}}\,

4lm^{2}=2(1+m^{2})\,

m^{4}+m^{2}-2=0,(m^{2}+2)(m^{2}-1)=0,m^{2}=-2,1,m=\pm 1\,

Therefore,the equations of the common tangent are y=\pm x\pm 2a=\pm (x+2a)\,


Main Page:Geometry:The Parabola