A) \[\left[ {{(x-1)}^{2}}+{{(y-1)}^{2}}+{{(z-1)}^{2}} \right]\left[ +\frac{4}{{{(x-y+z-1)}^{2}}} \right]=k\]
B) \[\left[ {{(x-1)}^{2}}+{{(y-1)}^{2}}+{{(z-1)}^{2}} \right]\left[ 1-\frac{4}{{{(x-y+z-1)}^{2}}} \right]=k\]
C) \[{{(x-1)}^{2}}+{{(y-1)}^{2}}+{{(z-1)}^{2}}+\frac{4}{{{(x-y+z-1)}^{2}}}=k\]
D) \[\frac{1}{{{(z-1)}^{2}}}+\frac{1}{{{(y-1)}^{2}}}+\frac{1}{{{(z-1)}^{2}}}+\frac{{{(x-y+z-1)}^{2}}}{4}=k\]
Correct Answer: A
Solution :
Let Q be\[(\alpha ,\beta ,\gamma )\]then \[P{{Q}^{2}}={{(\alpha -1)}^{2}}+{{(\beta -1)}^{2}}+{{(\gamma -1)}^{2}}=r_{2}^{2}\]where\[PQ={{r}_{2}}\] If \[PR={{r}_{1}}\] and \[l,m,n\] be the direction cosines of the line PR, then R is \[(1+l{{r}_{1}},1+m{{r}_{1}},1+n{{r}_{1}})\] R lies on the plane, so\[{{r}_{1}}=\frac{2}{l-m+n}\] Also, Q is\[(1+l{{r}_{2}},1=m{{r}_{2}}.1+n{{r}_{2}})\] \[\Rightarrow \]\[\frac{\alpha -1}{{{r}_{2}}}=l,\frac{\beta -1}{{{r}_{2}}}=m,\frac{\gamma -1}{{{r}_{2}}}=n\] \[\Rightarrow \]\[{{r}_{1}}=\frac{2{{r}_{2}}}{\alpha -\beta +\gamma -1}\] Now,\[r_{1}^{2}+r_{2}^{2}=k\] \[\Rightarrow \]\[r_{2}^{2}\left[ 1+\frac{4}{(\alpha -\beta +\gamma -1)} \right]=k\] Locus of Q is \[\left[ {{(x-1)}^{2}}+{{(y-1)}^{2}}+{{(z-1)}^{2}} \right]\left[ 1+\frac{4}{{{(x-y+z-1)}^{2}}} \right]=k\]
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