JEE Main & Advanced Sample Paper JEE Main Sample Paper-13

  • question_answer
    The locus of the mid-points of the chords of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] which are tangent to the ellipse \[\frac{{{x}^{2}}}{{{p}^{2}}}+\frac{{{y}^{2}}}{{{q}^{2}}}=1\] is

    A)  \[\left( \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}} \right)=\frac{{{p}^{2}}{{x}^{2}}}{{{a}^{4}}}+\frac{{{q}^{2}}{{y}^{2}}}{{{b}^{4}}}\]

    B)  \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{{{x}^{2}}}{{{p}^{2}}}+\frac{{{y}^{2}}}{{{q}^{2}}}\]

    C)  \[\left( \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}} \right)=\frac{{{p}^{2}}}{{{a}^{2}}}+\frac{{{q}^{2}}}{{{b}^{2}}}\]

    D)  \[\left( \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}} \right)=\frac{{{p}^{2}}}{{{q}^{2}}}+\frac{{{a}^{2}}}{{{b}^{2}}}\]

    Correct Answer: A

    Solution :

     Let the mid-point of any chord be (h, k). Then,          \[{{k}_{2}}\times {{10}^{3}}={{k}_{1}}\] \[+2{{H}_{2}}O+N{{H}_{4}}Cl\]   \[-OC{{H}_{3}}\] \[-I\]      \[OC{{H}_{3}}\] \[A{{l}_{2}}{{S}_{3}}+6{{H}_{2}}O\xrightarrow{\,}\,2Al{{(OH)}_{3}}+3{{H}_{2}}S\]              \[M{{g}^{2+}}\] \[A{{l}^{3+}}\] Condition for tangency between the line \[=2\times 4=8\] and ellipse is \[\therefore \] \[=\frac{1}{8}\times 100\]            \[AlC{{l}_{3}}\] Hence, locus of a point is \[HCl\] \[AlC{{l}_{3}}+3{{H}_{2}}O\xrightarrow{\,}\,Al{{(OH)}_{3}}+3HCl\]            \[(a=b=c\] \[\alpha =\beta =\gamma ={{90}^{o}}\] \[(a=b=c\]


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