Chapter 26

Statement \(-1:\) The point \(\mathrm{A}(3,1,6)\) is the mirror image of the point \(\mathrm{B}(1,3,4)\) in the plane \(x-y+z=5\). Statement \(-2:\) The plane \(x-y+z=5\) bisects the line segment joining \(\mathrm{A}(3,1,6)\) and \(\mathrm{B}(1,3,4)\). (a) Statement \(-1\) is true, Statement \(-2\) is true ; Statement 2 is not a correct explanation for Statement \(-1\). (b) Statement \(-1\) is true, Statement \(-2\) is false. (c) Statement \(-1\) is false, Statement \(-2\) is true. (d) Statement - 1 is true, Statement 2 is true; Statement \(-2\) is a correct explanation for Statement \(-1\).

Let the line \(\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}\) lie in the plane \(x+3 y-\alpha z+\beta=0 .\) Then \((\alpha, \beta)\) equals \(\quad\) [2009] (a) \((-6,7)\) (b) \((5,-15)\) (c) \((-5,5)\) (d) \((6,-17)\)

The line passing through the points \((5,1, a)\) and \((3, b, 1)\) crosses the yz-plane at the point \(\left(0, \frac{17}{2}, \frac{-13}{2}\right)\). Then \([2008]\) (a) \(a=2, b=8\) (b) \(a=4, b=6\) (c) \(a=6, b=4\) (d) \(a=8, b=2\)

Let \(L\) be the line of intersection of the planes \(2 x+3 y+z=\) 1 and \(x+3 y+2 z=2 .\) If \(L\) makes an angle \(\alpha\) with the positive \(x\)-axis, then \(\cos \alpha\) equals (a) 1 (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{\sqrt{3}}\) (d) \(\frac{1}{2}\).

The image of the point \((-1,3,4)\) in the plane \(x-2 y=0\) is \([2006]\) (a) \(\left(-\frac{17}{3},-\frac{19}{3}, 4\right)\) (b) \((15,11,4)\) (c) \(\left(-\frac{17}{3},-\frac{19}{3}, 1\right)\) (d) None of these

The distance between the line \(\vec{r}=2 \hat{i}-2 \hat{j}+3 \hat{k}+\lambda(i-j+4 k)\) and the plane \(\vec{r} \cdot(\hat{i}+5 \hat{j}+\hat{k})=5\) is (a) \(\frac{10}{9}\) (b) \(\frac{10}{3 \sqrt{3}}\) (c) \(\frac{3}{10}\) (d) \(\frac{10}{3}\)

If the angle \(\theta\) between the line \(\frac{x+1}{1}=\frac{y-1}{2}\) \(=\frac{z-2}{2}\) and the plane \(2 \mathrm{x}-\mathrm{y}+\sqrt{\lambda} \mathrm{z}+4=0\) is such that \(\sin \theta=\frac{1}{3}\) then the value of \(\lambda\) is \(\quad\) [2005] (a) \(\frac{5}{3}\) (b) \(\frac{-3}{5}\) (c) \(\frac{3}{4}\) (d) \(\frac{-4}{3}\)

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

The d.r. of normal to the plane through \((1,0,0),(0,1,0)\) which makes an angle \(\pi / 4\) with plane \(x+y=3\) are (a) \(1, \sqrt{2}, 1\) (b) \(1,1, \sqrt{2}\) (c) \(1,1,2\) (d) \(\sqrt{2}, 1,1\)

A plane which passes through the point \((3,2,0)\) and the line \(\frac{x-4}{1}=\frac{y-7}{5}=\frac{z-4}{4}\) is (a) \(x-y+z=1\) (b) \(x+y+z=5\) (c) \(x+2 y-z=1\) (d) \(2 x-y+z=5\)

If \((2,3,5)\) is one end of a diameter of the sphere \(x^{2}+y^{2}+z^{2}\) \(-6 x-12 y-2 z+20=0\), then the cooordinates of the other end of the diameter are (a) \((4,3,5)\) (b) \((4,3,-3)\) (c) \((4,9,-3)\) (d) \((4,-3,3)\).

The plane \(x+2 y-z=4\) cuts the sphere \(x^{2}+y^{2}+z^{2}-x+\) \(z-2=0\) in a circle of radius (a) 3 (b) 1 (c) 2 (d) \(\sqrt{2}\)

If the plane \(2 \mathrm{ax}-3 \mathrm{ay}+4 \mathrm{az}+6=0\) passes through the midpoint of the line joining the centres of the spheres \(x^{2}+y^{2}+z^{2}+6 x-8 y-2 z=13\) and \(x^{2}+y^{2}+z^{2}-10 x+4 y-2 z=8\) then a equals [2005] (a) \(-1\) (b) 1 (c) \(-2\) (d) 2

The intersection of the spheres \(x^{2}+y^{2}+z^{2}+7 x-2 y-z=13\) and \(x^{2}+y^{2}+z^{2}-3 x+3 y+4 z=8\) is the same as the intersection of one of the sphere and the plane [2004] (a) \(2 x-y-z=1\) (b) \(x-2 y-z=1\) (c) \(x-y-2 z=1\) (d) \(x-y-z=1\)

The radius of the circle in which the sphere \(x^{2}+y^{2}+z^{2}+2 x-2 y-4 z-19=0\) is cut by the plane \(x+2 y+2 z+7=0\) is [2003] (a) 4 (b) 1 (c) 2 (d) 3

The shortest distance from the plane \(12 x+4 y+3 z=327\) to the sphere \(x^{2}+y^{2}+z^{2}+4 x-2 y-6 z=155\) is [2003] (a) 39 (b) 26 (c) \(11 \frac{4}{13}\) (d) 13 .