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જો ઉપવલય $\frac{{{x^2}}}{{{a^2}}}\,\, + \;\,\frac{{{y^2}}}{{{b^2}}}\,\, = \,\,1$ નો કોઈપણ સ્પર્શક અક્ષો પર $h$ અને $k$ લંબાઈનો અંત:ખંડ કાપે, તો.....
$\frac{{{h^2}}}{{{a^2}}}\,\, + \,\,\frac{{{k^2}}}{{{b^2}}}\,\, = \,\,1$
$\frac{{{h^2}}}{{{a^2}}}\,\, + \,\,\frac{{{k^2}}}{{{b^2}}}\,\, = \,\,2$
$\frac{{{a^2}}}{{{h^2}}}\,\, + \,\,\frac{{{b^2}}}{{{k^2}}}\,\, = \,\,1$
$\frac{{{a^2}}}{{{h^2}}}\,\, + \,\,\frac{{{b^2}}}{{{k^2}}}\,\, = \,\,2$
Solution
ઉપવલય ને $\left( {{\text{a}}\,\,{\text{cos}}\,\theta ,\,b\,\,\sin \theta } \right)$ આગળનો સ્પર્શક:
$\frac{{\left( {a\,\,\cos \,\theta } \right)x}}{{{a^2}}}\,\, + \;\,\frac{{\left( {b\,\,\sin \,\,\theta } \right)\,\,y}}{{{b^2}}}\,\, = \,\,1\,$ અથવા
$\frac{x}{{\left( {a/\,\,\cos \,\,\theta } \right)}}\,\, + \;\,\frac{y}{{\left( {b/\,\sin \,\,\theta } \right)}}\,\, = \,\,1$
અંત : ખંડો ; $h\,\, = \,\,\frac{a}{{\cos \,\,\theta }},\,\,k\,\, = \,\,\frac{b}{{\sin \,\,\theta }}\,\, \Rightarrow \,\,\frac{{{a^2}}}{{{h^2}}}\,\, + \;\,\frac{{{b^2}}}{{{k^2}}}\,\, = \,\,1$
