A) \[\frac{1}{2},\,\,-2\]
B) \[-\frac{1}{2},\,\,2\]
C) \[-\frac{1}{2},\,\,-2\]
D) \[\frac{1}{2},\,\,2\]
Correct Answer: C
Solution :
We know that, two planes \[\overset{\to }{\mathop{\mathbf{r}}}\,\cdot \overset{\to }{\mathop{{{\mathbf{n}}_{\mathbf{1}}}}}\,={{d}_{1}}\] and \[\overset{\to }{\mathop{\mathbf{r}}}\,\cdot \overset{\to }{\mathop{{{\mathbf{n}}_{2}}}}\,={{d}_{2}}\] are parallel, if \[\overset{\to }{\mathop{{{\mathbf{n}}_{\mathbf{1}}}}}\,\] and \[\overset{\to }{\mathop{{{\mathbf{n}}_{\mathbf{2}}}}}\,\] are parallel \[ie,\] \[\overset{\to }{\mathop{{{\mathbf{n}}_{\mathbf{1}}}}}\,=t\overset{\to }{\mathop{{{\mathbf{n}}_{\mathbf{2}}}}}\,\] Thus, \[\overset{\to }{\mathop{\mathbf{r}}}\,(2\widehat{\mathbf{i}}-\lambda \widehat{\mathbf{j}}+\widehat{\mathbf{k}})=3\] and \[\overset{\to }{\mathop{\mathbf{r}}}\,\cdot (4\widehat{\mathbf{i}}+\widehat{\mathbf{j}}-\mu \widehat{\mathbf{k}})\] \[\therefore \] \[2\widehat{\mathbf{i}}-\lambda \widehat{\mathbf{j}}+\widehat{\mathbf{k}}=t(4\widehat{\mathbf{i}}+\widehat{\mathbf{j}}-\mu \widehat{\mathbf{k}})\] On comparing the coefficient of\[\widehat{\mathbf{i}},\,\,\widehat{\mathbf{j}},\,\,\widehat{\mathbf{k}}\], we get \[2=4t\Rightarrow t=\frac{1}{2}\] \[-\lambda =t\Rightarrow \lambda =-t=-\frac{1}{2}\] \[1=-\mu t\Rightarrow \mu =-\frac{1}{t}=-2\] Hence, \[\lambda =-1/2,\,\,\mu =-2\]
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