Geo5.1.21

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Find the equations of common tangents to the circle x^{2}+y^{2}=8\, and to the parabola y^{2}=16x\,

Any tangent to the parabola is

y=mx+4lm,mx-y+{\frac  {4}{m}}=0\,

This line is also tangent to the given circle

Hence

{\frac  {|{\frac  {4}{m}}|}{{\sqrt  {1+m^{2}}}}}={\sqrt  {8}}\,

16=8m^{2}(1+m^{2})\,

8m^{4}+8m^{2}-16=0\,

m^{4}+m^{2}-2=0\,

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

m^{2}=-2,1,\pm 1\,

Therefore,the equations of the common tangent are

y=\pm x\pm 4=\pm (x+4)\,

Main Page:Geometry:The Parabola