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

121243 Find the shortest distance between the lines
\(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \text { and } \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\)

1 \(2 \sqrt{3}\)
2 \(\sqrt{3}\)
3 \(4 \sqrt{3}\)
4 None of these
GUJCET-2011
Three Dimensional Geometry

121250 The shortest distance between the skew-lines \(\mathbf{r}=(-\hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{k}})+\mathbf{t}(2 \hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{j}}+\mathbf{6} \hat{\mathbf{k}})\) and
\(\mathbf{r}=(\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\mathbf{s}(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is

1 \(\frac{10}{\sqrt{17}}\)
2 \(\frac{22}{\sqrt{17}}\)
3 9
4 8
TS EAMCET 14.09.2020
Three Dimensional Geometry

121253 The equation of a plane containing the point \((1,-1,2)\) and perpendicular to the plane \(2 x+3 y-2 z=5\) and \(x+2 y-3 z=8\) is

1 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=7\)
2 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=5\)
3 \(\bar{r} \cdot(5 \hat{i}+4 \hat{j}+2 \hat{k})=5\)
4 \(\bar{r} \cdot(4 \hat{i}-5 \hat{j}+3 \hat{k})=15\)
MHT CET-2020
Three Dimensional Geometry

121259 If lines \(\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}\) intersect each other, then \(\lambda=\)

1 \(\frac{3}{2}\)
2 \(\frac{5}{2}\)
3 \(\frac{9}{2}\)
4 \(\frac{7}{2}\)
MHT CET-2019
Three Dimensional Geometry

121278 Find the equation of the plane which passes through the points \((0,1,2)\) and \((-1,0,3)\) and is perpendicular to the plane \(2 x+3 y+z=5\)

1 \(3 x-4 y+18 z+32=0\)
2 \(3 x+4 y-18 z+32=0\)
3 \(4 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+1=0\)
4 \(4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}+1=0\)
APEAPCET- 23.08.2021
NEET DIGITAL LIBRARY
Three Dimensional Geometry

121243 Find the shortest distance between the lines
\(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \text { and } \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\)

1 \(2 \sqrt{3}\)
2 \(\sqrt{3}\)
3 \(4 \sqrt{3}\)
4 None of these
GUJCET-2011
Three Dimensional Geometry

121250 The shortest distance between the skew-lines \(\mathbf{r}=(-\hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{k}})+\mathbf{t}(2 \hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{j}}+\mathbf{6} \hat{\mathbf{k}})\) and
\(\mathbf{r}=(\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\mathbf{s}(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is

1 \(\frac{10}{\sqrt{17}}\)
2 \(\frac{22}{\sqrt{17}}\)
3 9
4 8
TS EAMCET 14.09.2020
Three Dimensional Geometry

121253 The equation of a plane containing the point \((1,-1,2)\) and perpendicular to the plane \(2 x+3 y-2 z=5\) and \(x+2 y-3 z=8\) is

1 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=7\)
2 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=5\)
3 \(\bar{r} \cdot(5 \hat{i}+4 \hat{j}+2 \hat{k})=5\)
4 \(\bar{r} \cdot(4 \hat{i}-5 \hat{j}+3 \hat{k})=15\)
MHT CET-2020
Three Dimensional Geometry

121259 If lines \(\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}\) intersect each other, then \(\lambda=\)

1 \(\frac{3}{2}\)
2 \(\frac{5}{2}\)
3 \(\frac{9}{2}\)
4 \(\frac{7}{2}\)
MHT CET-2019
Three Dimensional Geometry

121278 Find the equation of the plane which passes through the points \((0,1,2)\) and \((-1,0,3)\) and is perpendicular to the plane \(2 x+3 y+z=5\)

1 \(3 x-4 y+18 z+32=0\)
2 \(3 x+4 y-18 z+32=0\)
3 \(4 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+1=0\)
4 \(4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}+1=0\)
APEAPCET- 23.08.2021
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Three Dimensional Geometry

121243 Find the shortest distance between the lines
\(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \text { and } \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\)

1 \(2 \sqrt{3}\)
2 \(\sqrt{3}\)
3 \(4 \sqrt{3}\)
4 None of these
GUJCET-2011
Three Dimensional Geometry

121250 The shortest distance between the skew-lines \(\mathbf{r}=(-\hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{k}})+\mathbf{t}(2 \hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{j}}+\mathbf{6} \hat{\mathbf{k}})\) and
\(\mathbf{r}=(\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\mathbf{s}(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is

1 \(\frac{10}{\sqrt{17}}\)
2 \(\frac{22}{\sqrt{17}}\)
3 9
4 8
TS EAMCET 14.09.2020
Three Dimensional Geometry

121253 The equation of a plane containing the point \((1,-1,2)\) and perpendicular to the plane \(2 x+3 y-2 z=5\) and \(x+2 y-3 z=8\) is

1 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=7\)
2 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=5\)
3 \(\bar{r} \cdot(5 \hat{i}+4 \hat{j}+2 \hat{k})=5\)
4 \(\bar{r} \cdot(4 \hat{i}-5 \hat{j}+3 \hat{k})=15\)
MHT CET-2020
Three Dimensional Geometry

121259 If lines \(\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}\) intersect each other, then \(\lambda=\)

1 \(\frac{3}{2}\)
2 \(\frac{5}{2}\)
3 \(\frac{9}{2}\)
4 \(\frac{7}{2}\)
MHT CET-2019
Three Dimensional Geometry

121278 Find the equation of the plane which passes through the points \((0,1,2)\) and \((-1,0,3)\) and is perpendicular to the plane \(2 x+3 y+z=5\)

1 \(3 x-4 y+18 z+32=0\)
2 \(3 x+4 y-18 z+32=0\)
3 \(4 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+1=0\)
4 \(4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}+1=0\)
APEAPCET- 23.08.2021
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Three Dimensional Geometry

121243 Find the shortest distance between the lines
\(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \text { and } \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\)

1 \(2 \sqrt{3}\)
2 \(\sqrt{3}\)
3 \(4 \sqrt{3}\)
4 None of these
GUJCET-2011
Three Dimensional Geometry

121250 The shortest distance between the skew-lines \(\mathbf{r}=(-\hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{k}})+\mathbf{t}(2 \hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{j}}+\mathbf{6} \hat{\mathbf{k}})\) and
\(\mathbf{r}=(\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\mathbf{s}(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is

1 \(\frac{10}{\sqrt{17}}\)
2 \(\frac{22}{\sqrt{17}}\)
3 9
4 8
TS EAMCET 14.09.2020
Three Dimensional Geometry

121253 The equation of a plane containing the point \((1,-1,2)\) and perpendicular to the plane \(2 x+3 y-2 z=5\) and \(x+2 y-3 z=8\) is

1 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=7\)
2 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=5\)
3 \(\bar{r} \cdot(5 \hat{i}+4 \hat{j}+2 \hat{k})=5\)
4 \(\bar{r} \cdot(4 \hat{i}-5 \hat{j}+3 \hat{k})=15\)
MHT CET-2020
Three Dimensional Geometry

121259 If lines \(\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}\) intersect each other, then \(\lambda=\)

1 \(\frac{3}{2}\)
2 \(\frac{5}{2}\)
3 \(\frac{9}{2}\)
4 \(\frac{7}{2}\)
MHT CET-2019
Three Dimensional Geometry

121278 Find the equation of the plane which passes through the points \((0,1,2)\) and \((-1,0,3)\) and is perpendicular to the plane \(2 x+3 y+z=5\)

1 \(3 x-4 y+18 z+32=0\)
2 \(3 x+4 y-18 z+32=0\)
3 \(4 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+1=0\)
4 \(4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}+1=0\)
APEAPCET- 23.08.2021
NEET DIGITAL LIBRARY
Three Dimensional Geometry

121243 Find the shortest distance between the lines
\(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \text { and } \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\)

1 \(2 \sqrt{3}\)
2 \(\sqrt{3}\)
3 \(4 \sqrt{3}\)
4 None of these
GUJCET-2011
Three Dimensional Geometry

121250 The shortest distance between the skew-lines \(\mathbf{r}=(-\hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{k}})+\mathbf{t}(2 \hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{j}}+\mathbf{6} \hat{\mathbf{k}})\) and
\(\mathbf{r}=(\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\mathbf{s}(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is

1 \(\frac{10}{\sqrt{17}}\)
2 \(\frac{22}{\sqrt{17}}\)
3 9
4 8
TS EAMCET 14.09.2020
Three Dimensional Geometry

121253 The equation of a plane containing the point \((1,-1,2)\) and perpendicular to the plane \(2 x+3 y-2 z=5\) and \(x+2 y-3 z=8\) is

1 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=7\)
2 \(\overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\hat{\mathrm{k}})=5\)
3 \(\bar{r} \cdot(5 \hat{i}+4 \hat{j}+2 \hat{k})=5\)
4 \(\bar{r} \cdot(4 \hat{i}-5 \hat{j}+3 \hat{k})=15\)
MHT CET-2020
Three Dimensional Geometry

121259 If lines \(\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}\) intersect each other, then \(\lambda=\)

1 \(\frac{3}{2}\)
2 \(\frac{5}{2}\)
3 \(\frac{9}{2}\)
4 \(\frac{7}{2}\)
MHT CET-2019
Three Dimensional Geometry

121278 Find the equation of the plane which passes through the points \((0,1,2)\) and \((-1,0,3)\) and is perpendicular to the plane \(2 x+3 y+z=5\)

1 \(3 x-4 y+18 z+32=0\)
2 \(3 x+4 y-18 z+32=0\)
3 \(4 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+1=0\)
4 \(4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}+1=0\)
APEAPCET- 23.08.2021