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If the line y=mx+c\, is a tangent to the parabola y^{2}=4a(x+a)\, prove that the condition is c=am+{\frac  {a}{m}}\,

Given line is y=mx+c\, and the parabola is y^{2}=4a(x+a)\,

Translate the origin to (-a,0). Let (x,y) be a point w.r.t the original axes.

Let (X,Y) be the same point w.r.t the new axes. Then x=X-a,y=Y+0=Y\,

Therefore the transformed equations of the given equations are


In the above equations, the line is tangent to the parabola,

(c-am)={\frac  {a}{m}}\,

c=am+{\frac  {a}{m}}\,

Therefore,if the given line is tangent to the parabola if c=am+{\frac  {a}{m}}\,

Main Page:Geometry:The Parabola