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જો ઉપવલય $x^2 + 2y^2 = 2$ શિરોબિંદુઓ સિવાયના બધા બિંદુઓથી સ્પર્શક દોરવામાં આવે તો બધા સ્પર્શકોના મધ્યબિંદુનો બિંદુપથ ............. થાય
$\frac{1}{{4{x^2}}} + \frac{1}{{2{y^2}}} = 1$
$\frac{{{x^2}}}{4} + \frac{{{y^2}}}{2} = 1$
$\frac{1}{{2{x^2}}} + \frac{1}{{4{y^2}}} = 1$
$\frac{{{x^2}}}{2} + \frac{{{y^2}}}{4} = 1$
Solution
equation of tangent is
$\frac{x}{a}\cos \theta + \frac{y}{b}\sin \theta = 1$
$A$ is $\left( {\frac{a}{{\cos \theta }},0} \right)$
$B$ is $\left( {0,\frac{b}{{\sin \theta }}} \right)$
Let $P\left( {h,k} \right)$ is mid point
$2h = \frac{a}{{\cos \theta }}$
$2k = \frac{b}{{\sin \theta }}$
${\cos ^2}\theta + {\sin ^2}\theta = 1$
$ \Rightarrow \frac{{{a^2}}}{{4{h^2}}} + \frac{{{b^2}}}{{4{k^2}}} = 1$
$ \Rightarrow \frac{2}{{4{x^2}}} + \frac{1}{{4{y^2}}} = 1$
$ \Rightarrow \frac{1}{{2{x^2}}} + \frac{1}{{4{y^2}}} = 1$
