Chapter 26

A plane which bisects the angle between the two given planes \(2 x-y+2 z-4=0\) and \(x+2 y+2 z-2=0\), passes through the point: [April 12, 2019 (II)] (a) \((1,-4,1)\) (b) \((1,4,-1)\) (c) \((2,4,1)\) (d) \((2,-4,1)\)

The length of the perpendicular drawn from the point ( 2 , 1,4 ) to the plane containing the lines \(\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+2 \hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-2 \hat{k})\) is: [April 12, 2019 (II)] (a) 3 (b) \(\frac{1}{3}\) (c) \(\sqrt{3}\) (d) \(\frac{1}{\sqrt{3}}\)

If the plane \(2 x-y+2 z+3=0\) has the distances \(\frac{1}{2}\) and \(\frac{2}{3}\) units from the planes \(4 x-2 y+4 z+\lambda=0\) and \(2 x-y+2 z+\) \(\mu=0\), respectively, then the maximum value of \(\lambda+\mu\) is equal to : [April 10, 2019 (II)] (a) 9 (b) 15 (c) 5 (d) 13

If the line, \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}\) meets the plane, \(x+2 y+3 z=15\) at a point \(P\), then the distance of \(P\) from the origin is: [April 09 2019I](a) \(\sqrt{5} / 2\) (b) \(2 \sqrt{5}\) (c) \(9 / 2\) (d) \(7 / 2\)

A plane passing through the points \((0,-1,0)\) and \((0,0,1)\) and making an angle \(\frac{\pi}{4}\) with the plane \(\mathrm{y}-\mathrm{z}+5=0\), also passes through the point: [April 09 2019I] (a) \((-\sqrt{2}, 1,-4)\) (b) \((\sqrt{2},-1,4)\) (c) \((-\sqrt{2},-1,-4)\) (d) \((\sqrt{2}, 1,4)\)

Let \(\mathrm{P}\) be the plane, which contains the line of intersection of the planes, \(x+y+z-6=0\) and \(2 x+3 y+z+5=0\) and it is perpendicular to the \(x y\)-plane. Then the distance of the point \((0,0,256)\) from P is equal to: [April 09, 2019 (II)] (a) \(17 / \sqrt{5}\) (b) \(63 \sqrt{5}\) (c) \(205 \sqrt{5}\) (d) \(11 / \sqrt{5}\)

The equation of a plane containing the line of intersection of the planes \(2 x-y-4=0\) and \(y+2 z-4=0\) and passing through the point \((1,1,0)\) is: (a) \(x-3 y-2 z=-2\) (b) \(2 x-z=2\) (c) \(x-y-z=0\) (d) \(x+3 y+z=4\)

The vector equation of the plane through the line of intersection of the planes \(x+y+z=1\) and \(2 x+3 y+4 z=5\) which is perpendicular to the plane \(x-y+z=0\) is: [April 08, 2019 (II)] (a) \(\vec{r} \times(\hat{i}-\hat{k})+2=0\) (b) \(\vec{r} \cdot(\hat{i}-\hat{k})-2=0\) (c) \(\vec{r} \times(\hat{i}+\hat{k})+2=0\) (d) \(\vec{r} \cdot(\hat{i}-\hat{k})+2=0\)

The perpendicular distance from the origin to the plane containing the two lines, \(\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}\) and \(\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}\), is: \(\quad\) [Jan. 12, 2019 (I)] (a) \(11 \sqrt{6}\) (b) \(11 / \sqrt{6}\) (c) 11 (d) \(6 \sqrt{11}\)

If an angle between the line, \(\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}\) and the plane, \(x-2 y-k x=3\) is \(\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\), then a value of \(k\) is [Jan. 12, 2019 (II)] (a) \(\sqrt{\frac{5}{3}}\) (b) \(\sqrt{\frac{3}{5}}\)(c) \(-\frac{3}{5}\) (d) \(-\frac{5}{3}\)

Let \(S\) be the set of all real values of \(\lambda\) such that a plane passing through the points \(\left(-\lambda^{2}, 1,1\right),\left(1,-\lambda^{2}, 1\right)\) and \(\left(1,1,-\lambda^{2}\right)\) also passes through the point- \((-1,-1,1)\). Then \(\mathrm{S}\) is equal to : [Jan. 12, 2019 (II)] (a) \(\\{\sqrt{3}\\}\) (b) \(\\{\sqrt{3},-\sqrt{3}\\}\) (c) \(\\{1,-1\\}\) (d) \(\\{3,-3\\}\)

The plane containing the line \(\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}\) and also containing its projection on the plane \(2 \mathrm{x}+3 \mathrm{y}-\mathrm{z}=5\), contains which one of the following points? [Jan. 11, 2019 (I)] (a) \((2,2,0)\) (b) \((-2,2,2)\) (c) \((0,-2,2)\) (d) \((2,0,-2)\)

The plane which bisects the line segment joining the points \((-3,-3,4)\) and \((3,7,6)\) at right angles, passes through which one of the following points? [Jan. 10, 2019 (II)] (a) \((-2,3,5)\) (b) \((4,-1,7)\) (c) \((2,1,3)\) (d) \((4,1,-2)\)

On which of the following lines lies the point of inter-section of the line, \(\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}\) and the plane, \(x+y+z=2 ? \quad\) [Jan. 10, 2019 (II)] (a) \(\frac{x+3}{3}=\frac{4-y}{3}=\frac{z+1}{-2}\) (b) \(\frac{x-4}{1}=\frac{y-5}{1}=\frac{z-5}{-1}\) (c) \(\frac{x-1}{1}=\frac{y-3}{2}=\frac{z+4}{-5}\) (d) \(\frac{x-2}{2}=\frac{y-3}{2}=\frac{z+3}{3}\)

The system of linear equations \(x+y+z=2\) \(2 x+3 y+2 z=5\) \(2 x+3 y+\left(a^{2}-1\right) z=a+1 \quad\) [Jan 09 2019I] (a) is inconsistent when \(\mathrm{a}=4\) (b) has a unique solution for \(|\mathrm{a}|=\sqrt{3}\) (c) has infinitely many solutions for \(\mathrm{a}=4\) (d) is inconsistent when \(|\mathrm{a}|=\sqrt{3}\)

The equation of the line passing through \((-4,3,1)\), parallel to the plane \(x+2 y-z-5=0\) and intersecting the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}\) is: \(\quad\) [Jan 09 2019I] (a) \(\frac{x-4}{2}=\frac{y+3}{1}=\frac{z+1}{4}\) (b) \(\frac{x+4}{1}=\frac{y-3}{1}=\frac{z-1}{3}\) (c) \(\frac{x+4}{3}=\frac{y-3}{-1}=\frac{z-1}{1}\) (d) \(\frac{x+4}{-1}=\frac{y-3}{1}=\frac{z-1}{1}\)

The plane through the intersection of the planes \(x+y+z=1\) and \(2 x+3 y-z+4=0\) and parallel to \(y\)-axis also passes through the point: \(\quad\) [Jan 09 2019I] (a) \((-3,0,-1)\) (b) \((-3,1,1)\) (c) \((3,3,-1)\) (d) \((3,2,1)\)

The equation of the plane containing the straight line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\) and perpendicular to the plane containing the straight lines \(\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\) and \(\frac{x}{4}=\frac{y}{2}=\frac{z}{3}\) is: (II)] [Jan. 09, 2019 (a) \(x-2 y+z=0\) (b) \(3 x+2 y-3 z=0\) (c) \(x+2 y-2 z=0\) (d) \(5 x+2 y-4 z=0\)

If \(L_{1}\) is the line of intersection of the planes \(2 x-2 y+3 z-2=0, x-y+z+1=0\) and \(L_{2}\) is the line of intersection of the planes \(x+2 y-z-3=0\), \(3 x-y+2 z-1=0\), then the distance of the origin from the plane, containing the lines \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\), is : [2018] (a) \(\frac{1}{3 \sqrt{2}}\) (b) \(\frac{1}{2 \sqrt{2}}\) (c) \(\frac{1}{\sqrt{2}}\) (d) \(\frac{1}{4 \sqrt{2}}\)

A variable plane passes through a fixed point \((3,2,1)\) and meets \(x, y\) and \(z\) axes at \(A, B\) and \(C\) respectively. A plane is drawn parallel to \(y z-p\) lane through \(A\), a second plane is drawn parallel \(z x-\) plane through \(B\) and a third plane is drawn parallel to \(x y-\) plane through \(C\). Then the locus of the point of intersection of these three planes, is [Online April 15, 2018] (a) \(x+y+z=6\) (b) \(\frac{x}{3}+\frac{y}{2}+\frac{z}{1}=1\) (c) \(\frac{3}{x}+\frac{2}{y}+\frac{1}{z}=1\) (d) \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{11}{6}\)

A plane bisects the line segment joining the points \((1,2,3)\) and \((-3,4,5)\) at right angles. Then this plane also passes through the point. \(\quad\) Online April 15, 2018] (a) \((-3,2,1)\) (b) \((3,2,1)\) (c) \((1,2,-3)\) (d) \((-1,2,3)\)

If the image of the point \(\mathrm{P}(1,-2,3)\) in the plane, \(2 \mathrm{x}+3 \mathrm{y}-4 \mathrm{z}+22=0\) measured parallel to line, \(\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{4}=\frac{\mathrm{z}}{5}\) is \(Q\), then \(P Q\) is equal to : [2017] (a) \(6 \sqrt{5}\) (b) \(3 \sqrt{5}\) (c) \(2 \sqrt{42}\) (d) \(\sqrt{42}\)

The distance of the point \((1,3,-7)\) from the plane passing through the point \((1,-1,-1)\), having normal perpendicular to both the lines \(\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}\) and \(\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}\), is (a) \(\frac{10}{\sqrt{74}}\) (b) \(\frac{20}{\sqrt{74}}\) (c) \(\frac{10}{\sqrt{83}}\) (d) \(\frac{5}{\sqrt{83}}\)

If \(x=a, y=b, z=c\) is a solution of the system of linear equations [Online April9, 2017] \(x+8 y+7 z=0\) \(9 x+2 y+3 z=0\) \(x+y+z=0\) such that the point \((a, b, c)\) lies on the plane \(x+2 y+z=6\), then \(2 a+b+c\) equals: (a) \(-1\) (b) 0 (c) 1 (d) 2

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\), then the locus of the centroid of \(\Delta \mathrm{ABC}\) is: \([\) Online April 9, 2017] (a) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=1\) (b) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=3\) (c) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{9}\) (d) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=9\)

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\), then the locus of the centroid of \(\Delta \mathrm{ABC}\) is: \([\) Online April 9, 2017] (a) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=1\) (b) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=3\) (c) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{9}\) (d) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=9\)

The coordinates of the foot of the perpendicular from the point \((1,-2,1)\) on the plane containing the lines, \(\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}\) and \(\frac{x-1}{3}=\frac{y-2}{5}=\frac{z-3}{7}\), is: \(\quad\) Online April 8, 2017] (a) \((2,-4,2)\) (b) \((-1,2,-1)\) (c) \((0,0,0)\) (d) \((1,1,1)\)

The line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})=1\) and \(\begin{array}{ll}\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}) & =2 \text {, is : } & \text { [Online April 8, 2017] }\end{array}\) (a) \(\frac{x-\frac{4}{7}}{-2}=\frac{y}{7}=\frac{z-\frac{5}{7}}{13}\) (b) \(\frac{x-\frac{4}{7}}{2}=\frac{y}{-7}=\frac{z+\frac{5}{7}}{13}\) (c) \(\frac{x-\frac{6}{13}}{2}=\frac{y-\frac{5}{13}}{-7}=\frac{z}{-13}\) (d) \(\frac{x-\frac{6}{13}}{2}=\frac{y-\frac{5}{13}}{7}=\frac{z}{-13}\)

If the line, \(\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}\) lies in theplane, \(l x+m y-z=\) 9, then \(\mathrm{l}^{2}+\mathrm{m}^{2}\) is equal to : (a) 5 (b) 2 (c) 26 (d) 18

The distance of the point \((1,-5,9)\) from the plane \(x-y+z=\) 5 measured along the line \(x=y=z\) is : [2016] (a) \(\frac{10}{\sqrt{3}}\) (b) \(\frac{20}{3}\) (c) \(3 \sqrt{10}\) (d) \(10 \sqrt{3}\)

The distance of the point \((1,-5,9)\) from the plane \(x-y+z=\) 5 measured along the line \(x=y=z\) is : [2016] (a) \(\frac{10}{\sqrt{3}}\) (b) \(\frac{20}{3}\) (c) \(3 \sqrt{10}\) (d) \(10 \sqrt{3}\)

The distance of the point \((1,0,2)\) from the point of intersection of the line \(\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}\) and the plane \(x-y+z=16\), is \([2015]\) (a) \(3 \sqrt{21}\) (b) 13 (c) \(2 \sqrt{14}\) (d) 8

A plane containing the point \((3,2,0)\) and the line \(\frac{x-1}{1}=\frac{y-2}{5}=\frac{z-3}{4}\) also contains the point: [Online April 11, 2015] (a) \((0,3,1)\) (b) \((0,7,-10)\) (c) \((0,-3,1)\) (d) \(0,7,10\)

If the points \((1,1, \lambda)\) and \((-3,0,1)\) are equidistant from the plane, \(3 x+4 y-12 z+13=0\), then \(\lambda\) satisfies the equation: \(\quad\) [Online April 10, 2015] (a) \(3 x^{2}+10 x-13=0\) (b) \(3 x^{2}-10 x+21=0\) (c) \(3 x^{2}-10 x+7=0\) (d) \(3 x^{2}+10 x-7=0\)

The image of the line \(\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}\) in the plane \(2 x-y+z+3=0\) is the line: (a) \(\frac{x-3}{3}=\frac{y+5}{1}=\frac{z-2}{-5}\) (b) \(\frac{x-3}{-3}=\frac{y+5}{-1}=\frac{z-2}{5}\) (c) \(\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}\) (d) \(\frac{x+3}{-3}=\frac{y-5}{-1}=\frac{z+2}{5}\)

If the angle between the line \(2(x+1)=y=z+4\) and the plane \(2 \mathrm{x}-\mathrm{y}+\sqrt{\lambda} \mathrm{z}+4=0\) is \(\frac{\pi}{6}\), then the value of \(\lambda\) is: [Online April 19, 2014] (a) \(\frac{135}{7}\) (b) \(\frac{45}{11}\) (c) \(\frac{45}{7}\) (d) \(\frac{135}{11}\)

If the distance between planes, \(4 x-2 y-4 z+1=0\) and \(4 x-2 y-4 z+d=0\) is 7 , then dis: [Online April 12, 2014] (a) 41 or \(-42\) (b) 42 or \(-43\) (c) \(-41\) or 43 (d) \(-42\) or 44

A symmetrical form of the line of intersection of the planes \(x=a y+b\) and \(z=c y+d\) is \(\quad\) [Online April 12, 2014] (a) \(\frac{x-b}{a}=\frac{y-1}{1}=\frac{z-d}{c}\) (b) \(\frac{x-b-a}{a}=\frac{y-1}{1}=\frac{z-d-c}{c}\) (c) \(\frac{x-a}{b}=\frac{y-0}{1}=\frac{z-c}{d}\) (d) \(\frac{x-b-a}{b}=\frac{y-1}{0}=\frac{z-d-c}{d}\)

The plane containing the line \(\frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}-2}{2}=\frac{\mathrm{z}-3}{3}\) and parallel to the line \(\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{1}=\frac{\mathrm{z}}{4}\) passes through the point: [Online April 11, 2014] (a) \((1,-2,5)\) (b) \((1,0,5)\) (c) \((0,3,-5)\) (d) \((-1,-3,0)\)

Equation of the plane which passes through the point of intersection of lines \(\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}\) and \(\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) and has the largest distance from the origin is: [Online April 9, 2014] (a) \(7 x+2 y+4 z=54\) (b) \(3 x+4 y+5 z=49\) (c) \(4 x+3 y+5 z=50\) (d) \(5 x+4 y+3 z=57\)

Distance between two parallel planes \(2 x+y+2 z=8\) and \(4 x+2 y+4 z+5=0\) is (a) \(\frac{3}{2}\) (b) \(\frac{5}{2}\) (c) \(\frac{7}{2}\) (d) \(\frac{9}{2}\)

The equation of a plane through the line of intersection of the planes \(x+2 y=3, y-2 z+1=0\), and perpendicular to the first plane is : [Online April 25, 2013] (a) \(2 x-y-10 z=9\) (b) \(2 x-y+7 z=11\) (c) \(2 x-y+10 z=11\) (d) \(2 x-y-9 z=10\)

A vector \(\vec{n}\) is inclined to \(x\)-axis at \(45^{\circ}\), to \(y\)-axis at \(60^{\circ}\) and at an acute angle to \(z\)-axis. If \(\vec{n}\) is a normal to a plane passing through the point \((\sqrt{2},-1,1)\) then the equation of the plane is: \([\) Online April 9, 2013] (a) \(4 \sqrt{2} x+7 y+z-2\) (b) \(2 x+y+2 z=2 \sqrt{2}+1\) (c) \(3 \sqrt{2} x-4 y-3 z=7\) (d) \(\sqrt{2} x-y-z=2\)

A equation of a plane parallel to the plane \(x-2 y+2 z-5=0\) and at a unit distance from the origin is : [2012] (a) \(x-2 y+2 z-3=0\) (b) \(x-2 y+2 z+1=0\) (c) \(x-2 y+2 z-1=0\) (d) \(x-2 y+2 z+5=0\)

The equation of a plane containing the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\) and the point \((0,7,-7)\) is \(\begin{array}{ll}\text { (a) } x+y+z=0 & \text { [Online May 26, 2012] }\end{array}\) (b) \(x+2 y+z=21\) (c) \(3 x-2 y+5 z+35=0\) (d) \(3 x+2 y+5 z+21=0\)

Consider the following planes \(P: x+y-2 z+7=0\) \(Q: x+y+2 z+2=0\) \(R: 3 x+3 y-6 z-11=0 \quad\) [Online May 26, 2012] (a) \(P\) and \(R\) are perpendicular (b) \(Q\) and \(R\) are perpendicular (c) \(P\) and \(Q\) are parallel (d) \(P\) and \(R\) are parallel

If the three planes \(x=5,2 x-5 a y+3 z-2=0\) and \(3 b x+y-3 z=0\) contain a common line, then \((a, b)\) is equal to [Online May 19, 2012] (a) \(\left(\frac{8}{15},-\frac{1}{5}\right)\) (b) \(\left(\frac{1}{5},-\frac{8}{15}\right)\) (c) \(\left(-\frac{8}{15}, \frac{1}{5}\right)\) (d) \(\left(-\frac{1}{5}, \frac{8}{15}\right)\)

The values of a for which the two points \((1, a, 1)\) and \((-3,0, a)\) lie on the opposite sides of the plane \(3 x+4 y-\) \(12 z+13=0\), satisfy \(\quad\) [Online May 7, 2012] (a) \(0

The distance of the point \((1,-5,9)\) from the plane \(x-y+\) \(z=5\) measured along a straight \(x=y=z\) is \(\quad\) [2011RS] (a) \(10 \sqrt{3}\) (b) \(5 \sqrt{3}\) (c) \(3 \sqrt{10}\) (d) \(3 \sqrt{5}\)

If the angle between the line \(x=\frac{y-1}{2}=\frac{z-3}{\lambda}\) and the plane \(x+2 y+3 z=4\) is \(\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)\), then \(\lambda\) equals [2011](a) \(\frac{3}{2}\) (b) \(\frac{2}{5}\) (c) \(\frac{5}{3}\) (d) \(\frac{2}{3}\)