Question

Then the correct matching is

List - I	List - II
A: The coordinates of the mid point of the line joining $(-1,-1,1)$ and $(-1,1,-1)$	1) $(-2,1,1)$
B: The coordinates of the point which divides the line segment joining $($2,3,1 $)$ and $(5,0,4)$ in the ratio1:2	2) $(-1,0,0)$
$C$: The points and $P(2,1,-3)$ are three vertices of a parallelogram $PQRS$, the fourth vertex	3) $(\frac{13}{3},\frac{-11}{3},6)$
$D$: The vertices of a triangle are $(7,-4,7)$ , $(1,-6,10)$ and $(5,-1,1)$ . centroid of the triangle	4) $($3, 2, 2 $)$

• A. $A-2,B-4,C-1,D-3$
• B. $A-1,B-2,C-3,D-4$
• C. $A-2,B-3,C-1,D-4$
• D. $A-1,B-4,C-3,D-2$

Answer 1

The correct option is A $A-2,B-4,C-1,D-3$

A: The co-ordinate of the mid point of the line joining $(-1,-1,1)$ and $(-1,1,-1)$

from mid point formula $(\frac{-1-1}{2},\frac{-1+1}{2},\frac{1-1}{2})$

$=(-1,0,0)$

$A\to 2$

B: co-ordinates of the point which divides the line segment Joining

$(2,3,1)$ and $(5,0,4)$ in ratio $1:2$

so from section formula $(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2},\frac{m_1{z}_2+m_2{z}_1}{m_1+m_2})$

Here $m_1=1,m_2=2$

so $=(\frac{1\times +2\times 2}{3},\frac{1\left(0\right)+2\left(3\right)}{3},\frac{1\left(4\right)+2\left(1\right)}{3})$

$=(3,2,2)$

$B\to 4$

C: The point $(2,3,1),(5,0,4)$ and $(2,1,-3)$ and $A(x,y,z)$ vertices of a parallelogram $PQRS$ then fourth vertex

we know that the diagonals of parallelogram bisect each other have they mid point of $PR=$ mid point of $QS$ .... [Ref. image]

$\Rightarrow (\frac{2+5}{2},\frac{1}{2},\frac{1}{2})=(\frac{x+2}{2},\frac{y+3}{2},\frac{z+1}{2})$

$\Rightarrow \frac{x+2}{2}=\frac{7}{2}\Rightarrow x=5$

$\Rightarrow \frac{y+3}{2}=\frac{-1}{2}\Rightarrow y=-2$

$\Rightarrow \frac{z+1}{2}=\frac{1}{2}\Rightarrow z=0$

D: The vertixes of a triangle $ax$ $(7,-4,7),(1,-6,10)$ and $(5,-1,1)$ Centroid of triangle

Then centroid $=(\frac{7+1+5}{3},\frac{-4-6-1}{3},\frac{7+10+1}{3})$

$=(\frac{13}{3},\frac{-11}{3},6)$

$D\to 3$

Hence correct option is A