Column-I		Column-II
A)	The coordinates of a point on the line $x=4y+5,z=3y-6$ at a distance 3 from the point $(5,3,-6)$ is/are	p)	$\left(-1,-2,0\right)$
B)	The plane containing the lines $\frac{x-2}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ and parallel to $\widehat{i}+4\widehat{j}+7\widehat{k}$ has the point	q)	$\left(5,0,-6\right)$
C)	A line passes through two points $A\left(2,-3,-1\right)$ and $B\left(8,-1,2\right)$ The coordinates of a point on this line  nearer  to the origin and at a distance of  14  units form A is/are	r)	$\left(2,5,7\right)$
D)	The coordinates of the foot of the perpendicular from the point $\left(3,-1,11\right)$ on the line  $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ is/are	s)	$\left(14,1,5\right)$

 

A)  The given line is  $x=4y+5,zZ=3y=6$ or 

$\frac{x-5}{4}=y,\frac{z+6}{3}=y$

$\frac{x-5}{4}=\frac{y}{1}=\frac{z+6}{3}=\lambda \left(say\right)$ any point on the line is of the form $\left(4\lambda +5\lambda .3\lambda -6\right)$ and $\left(5.3,-6\right)$ is 3 units (given),

$26{\lambda }^2-6\lambda =0$ or $\lambda =0.3/13.$

The point is  $\left(5,0,-6\right)$

B)   The equation of the plane containing the lines $\frac{x-2}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ and parallel to $\hat{i}+4\hat{j}+7\hat{k}$ is $\left|\begin{array}{l}x-2y+3z+5\\ 1 47\\ 3 57\end{array}\right|=0$

    C)   The line passing through points. The line passing through points $A\left(2,-3,-1\right)$  and 

$B(8,-1,2)$ is $\frac{x-2}{8-2}=\frac{y+3}{-1+3}=\frac{z+1}{2+1}$

or $\frac{x-2}{6}=\frac{y+3}{-1+3}=\frac{z+3}{-1+3}=\frac{z+1}{Z+1}$

Any point on this line is of the form $P\left(6\lambda +2.2\lambda -3.3\lambda -1\right)$ , Whose distance from point $A\left(2,-3,-1\right)$is 14 units..

$\therefore PA=14orP{A}^2={\left(14\right)}^2$ or $4{\lambda }^2=196or{\lambda }^2=4or\lambda =\pm 2$

$\therefore$The requited points are $\left(14,1,5\right)$  and  $\left(10,-7,-7\right)$

The point nearer to the origin is  $\left(14,1,5\right)$

    D)   Any point on line $AB,\frac{x}{2}=\frac{x-2}{3}=\frac{z-3}{4}=\lambda .$ 
          Is  $M\left(2\lambda ,3\lambda +2,4\lambda -3\lambda \right).$

 
Therefore, The direction ratios of PM are $2\lambda -3,3\lambda +3$  and  $4\lambda -8,$ But  $PM\perp AB$

$4\lambda -6+9\lambda +9+26\lambda -32=0$

$\Rightarrow 29\lambda -29=0\Rightarrow \lambda =0$