Equation of a Line, Sphere, and a Plane in Different Forms
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Three Dimensional Geometry

121285 If \((2,-1,3)\) is the foot of the perpendicular drawn the origin to the plane, then the equation of the plane is

1 \(2 x+y-3 z+6=0\)
2 \(2 x-y+3 z-14=0\)
3 \(2 x-y+3 z-13=0\)
4 \(2 x-y+3 z-10=0\)
AP EAMCET-2004
Three Dimensional Geometry

121246 The shortest distance between the lines
\(\overrightarrow{\mathbf{r}}=(\mathbf{4} \hat{\mathbf{i}}-\hat{\mathbf{j}})+\lambda(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\mathbf{3} \hat{\mathbf{k}})\) and
\(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})+\boldsymbol{\mu}(\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}-\mathbf{5} \hat{\mathbf{k}})\) is

1 \(\frac{2}{\sqrt{5}}\)
2 \(\frac{2}{5}\)
3 \(\frac{6}{\sqrt{5}}\)
4 \(\frac{1}{\sqrt{5}}\)
AMU-2017
Three Dimensional Geometry

121268 The equation of a straight line parallel to the \(x\)-axis is given by

1 \(\frac{x-a}{1}=\frac{y-b}{1}=\frac{z-c}{1}\)
2 \(\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}\)
3 \(\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{1}\)
4 \(\frac{\mathrm{x}-\mathrm{a}}{1}=\frac{\mathrm{y}-\mathrm{b}}{0}=\frac{\mathrm{z}-\mathrm{c}}{0}\)
CG PET- 2013
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Three Dimensional Geometry

121285 If \((2,-1,3)\) is the foot of the perpendicular drawn the origin to the plane, then the equation of the plane is

1 \(2 x+y-3 z+6=0\)
2 \(2 x-y+3 z-14=0\)
3 \(2 x-y+3 z-13=0\)
4 \(2 x-y+3 z-10=0\)
AP EAMCET-2004
Three Dimensional Geometry

121246 The shortest distance between the lines
\(\overrightarrow{\mathbf{r}}=(\mathbf{4} \hat{\mathbf{i}}-\hat{\mathbf{j}})+\lambda(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\mathbf{3} \hat{\mathbf{k}})\) and
\(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})+\boldsymbol{\mu}(\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}-\mathbf{5} \hat{\mathbf{k}})\) is

1 \(\frac{2}{\sqrt{5}}\)
2 \(\frac{2}{5}\)
3 \(\frac{6}{\sqrt{5}}\)
4 \(\frac{1}{\sqrt{5}}\)
AMU-2017
Three Dimensional Geometry

121268 The equation of a straight line parallel to the \(x\)-axis is given by

1 \(\frac{x-a}{1}=\frac{y-b}{1}=\frac{z-c}{1}\)
2 \(\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}\)
3 \(\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{1}\)
4 \(\frac{\mathrm{x}-\mathrm{a}}{1}=\frac{\mathrm{y}-\mathrm{b}}{0}=\frac{\mathrm{z}-\mathrm{c}}{0}\)
CG PET- 2013
NEET DIGITAL LIBRARY
Three Dimensional Geometry

121285 If \((2,-1,3)\) is the foot of the perpendicular drawn the origin to the plane, then the equation of the plane is

1 \(2 x+y-3 z+6=0\)
2 \(2 x-y+3 z-14=0\)
3 \(2 x-y+3 z-13=0\)
4 \(2 x-y+3 z-10=0\)
AP EAMCET-2004
Three Dimensional Geometry

121246 The shortest distance between the lines
\(\overrightarrow{\mathbf{r}}=(\mathbf{4} \hat{\mathbf{i}}-\hat{\mathbf{j}})+\lambda(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\mathbf{3} \hat{\mathbf{k}})\) and
\(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})+\boldsymbol{\mu}(\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}-\mathbf{5} \hat{\mathbf{k}})\) is

1 \(\frac{2}{\sqrt{5}}\)
2 \(\frac{2}{5}\)
3 \(\frac{6}{\sqrt{5}}\)
4 \(\frac{1}{\sqrt{5}}\)
AMU-2017
Three Dimensional Geometry

121268 The equation of a straight line parallel to the \(x\)-axis is given by

1 \(\frac{x-a}{1}=\frac{y-b}{1}=\frac{z-c}{1}\)
2 \(\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}\)
3 \(\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{1}\)
4 \(\frac{\mathrm{x}-\mathrm{a}}{1}=\frac{\mathrm{y}-\mathrm{b}}{0}=\frac{\mathrm{z}-\mathrm{c}}{0}\)
CG PET- 2013
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