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KVPY Sample Paper KVPY Stream-SX Model Paper-17

  • question_answer
    \[P(a,b)\]is a points in the first quadrant. Circles are drawn through P touching the coordinate axes, such that the length of common chord of these circle is maximum. The possible values of \[a/b\]is

    A) \[3\pm 3\sqrt{2}\]

    B) \[3+2\sqrt{3}\]

    C) \[3-2\sqrt{3}\]

    D) \[3\pm 2\sqrt{2}\]

    Correct Answer: D

    Solution :

    Let \[r\]be the radius of the circle. Its equation is \[{{x}^{2}}+{{y}^{2}}-2r(x+y)+{{r}^{2}}=0.\]since it passes through \[P(a,b)\]
    \[{{a}^{2}}+{{b}^{2}}-2r(a+b)+{{r}^{2}}=0\]
    Solving \[{{r}_{1}}=a+b\sqrt{2ab}\]                              ? (1)
    \[{{r}_{2}}=a+b-\sqrt{2ab}\]
    Now, the equations of two circle are  \[{{x}^{2}}+{{y}^{2}}-2{{r}_{1}}(x+y)+r_{1}^{2}=0\] and \[{{x}^{2}}+{{y}^{2}}-2{{r}_{2}}(x+y)+r_{2}^{2}=0\] The common chord is \[{{S}_{1}}-{{S}_{2}}=0\] \[\Rightarrow \]\[2({{r}_{2}}-{{r}_{1}})(x+y)+r_{1}^{2}-r_{2}^{2}=0\] \[\Rightarrow \]\[2(x+y)+{{r}_{1}}+{{r}_{2}}\] For maximum length of the common chord, it must pass through the centre of the smaller circle\[({{r}_{2}},{{r}_{2}})\], so  \[4{{r}_{2}}={{r}_{1}}+{{r}_{2}}\Rightarrow \frac{{{r}_{1}}}{{{r}_{2}}}=3\] \[\Rightarrow \]\[\frac{a+b+\sqrt{2ab}}{a+b-\sqrt{2ab}}=3\] \[\Rightarrow \]\[2(a+b)=4\sqrt{2ab}\] \[\Rightarrow \]\[{{(a+b)}^{2}}=8ab\] \[\Rightarrow \]\[{{a}^{2}}-6ab+{{b}^{2}}=0\] \[\Rightarrow \]\[a=\frac{6b\pm \sqrt{36{{b}^{2}}-4{{b}^{2}}}}{2}=(3\pm 2\sqrt{2})b\] \[\Rightarrow \frac{a}{b}=3\pm 2\sqrt{2}\]


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