Geo5.2.39

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Prove that the pair of tangents drawn to 9x^{2}+16y^{2}=144\, are perpendicular to eachother.

Given S\equiv 9x^{2}+16y^{2}=144\, and Let the point be (x_{1},y_{1})\,

S_{1}\equiv 9xx_{1}+16yy_{1}-144=0\,

S_{{11}}\equiv 9x_{1}^{{2}}+16y_{1}^{{2}}-144\,

Therefore,equation to the pair of tangents is S_{1}^{{2}}=SS_{{11}}\,

(9xx_{1}+16yy_{1}-144)^{2}=(9x^{2}+16y^{2}-144)(9x_{1}^{{2}}+16y_{1}^{{2}}-144)\,

Separating the coefficients of x^{2}\, and y^{2}\,

81x_{1}^{{2}}+256y_{1}^{{2}}-81x_{1}^{{2}}-256y_{1}^{{2}}\,

From the above,the sum of the coefficients of squares of x and y is zero,which is the condition for the two lines to be perpendicular.

Hence the two tangents drawn on the given ellipse are perpendicular to each other.


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