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Find the value of k if the line x+ky-5=0\, is a tangent to the ellipse 4x^{2}+9y^{2}=20\,

Given tangent is x+ky-5=0\,

Given ellipse is 4x^{2}+9y^{2}=20\,

The condition for a line to be tangent to an ellipse is a^{2}l^{2}+b^{2}m^{2}=n^{2}\,

From the given equations l=1,m=k,n=-5,a^{2}=5,b^{2}={\frac  {20}{9}}\,

Substituting these values in the condition of tangency,we get

5(1)^{2}+{\frac  {20}{9}}(k^{2})=(-5)^{2}\,

45+20k^{2}=225,20k^{2}=180,k^{2}=9,k=\pm 3\,

Main Page:Geometry:The Ellipse