Geo5.2.17

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Find the equations to the tangents to the ellipse x^{2}+2y^{2}=3\, drawn from the point (1,2)

Any line through (1,2) with slope m is y-2=m(x-1),y=mx+(2-m)\,

This is a tangent to the given ellipse {\frac  {x^{2}}{3}}+{\frac  {y^{2}}{{\frac  {3}{2}}}}=1\,

Condition for tangency is c^{2}=a^{2}m^{2}+b^{2}\,

(2-m)^{2}=3m^{2}+{\frac  {3}{2}}\,

4m^{2}+8m-5=0,(2m+5)(2m-1)=0\,

m={\frac  {1}{2}},{\frac  {-5}{2}}\,

By substituting the values of m in the tangent equation,we get

Therefore,the equations of the tangents are x-2y+3=0,5x+2y-9=0\,


[Main page]]:Geometry:The Ellipse