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JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Critical Thinking

  • question_answer
    If \[{{P}_{1}}\] and \[{{P}_{2}}\] are the lengths of the perpendiculars from the points (2,3,4) and (1,1,4) respectively from the plane \[3x-6y+2z+11=0\], then \[{{P}_{1}}\] and \[{{P}_{2}}\] are the roots of the equation

    A) \[{{P}^{2}}-23P+7=0\]

    B) \[7{{P}^{2}}-23P+16=0\]           

    C) \[{{P}^{2}}-17P+16=0\]

    D) \[{{P}^{2}}-16P+7=0\]

    Correct Answer: B

    Solution :

    • We have,   \[{{P}_{1}}=\left| \,\frac{3\times 2-6\times 3+2\times 4+11}{\sqrt{{{3}^{2}}+{{(-6)}^{2}}+{{(2)}^{2}}}}\, \right|=1\]                                    
    • \[{{P}_{2}}=\left| \frac{3\times 1-6\times 1+2\times 4+11}{\sqrt{{{3}^{2}}+{{(-6)}^{2}}+{{(2)}^{2}}}} \right|=\frac{16}{7}\]                                
    • So, equation whose roots are \[{{P}_{1}}\]  and \[{{P}_{2}}\] is,                                                                      
    • \[7{{P}^{2}}-23P+16=0\].


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