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Show that the condition for a straight line y=mx+c\, be a tangent to the ellipse{\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\, is c^{2}=a^{2}m^{2}+b^{2}\,

The x coordinates of the points of intersection of the line y=mx+c\, and the

ellipse are given by


The line will touch the ellipse if the two points are coincident i.e if the roots of the equation above are equal.

Discriminant of the equation is zero.



c=\pm {\sqrt  {a^{2}m^{2}+b^{2}}}\,

Main Page:Geometry:The Ellipse