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J & K CET Engineering J and K - CET Engineering Solved Paper-2009

  • question_answer
    If \[\vec{a}\] is a constant vector and p is a real constant with \[|\vec{a}{{|}^{2}}>p,\] then the locus of a point with position vector \[|\vec{r}|\] such that \[|\vec{r}{{|}^{2}}-2\,\,\vec{r}.\,\vec{a}\,+\,\,p=O\]is

    A)  a sphere       

    B)  an ellipse

    C)  a circle          

    D)  a plane

    Correct Answer: A

    Solution :

    Let \[\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}\] and  \[\vec{a}=a\hat{i}+b\hat{j}+c\hat{k},\] where \[a,\,\,\,b,\,\,\,c\] are constant. Now, \[|\vec{r}{{|}^{2}}-2\vec{r}.\vec{a}+p=0\] \[\Rightarrow \]  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2(ax+by+cz)+p=0\] Which represent a sphere, where  radius \[=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-p}=+ve\] \[[\because \,\,|\vec{a}|>p|\]


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