Geo5.2.12

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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

(a^{2}m^{2}+b^{2})x^{2}+2a^{2}cmx+a^{2}(c^{2}-b^{2})=0\,

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.

4a^{4}m^{2}c^{2}-4(a^{2}m^{2}+b^{2})(a^{2})(c^{2}-b^{2})=0\,

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

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


Main Page:Geometry:The Ellipse