Equation of a Line Passing through Two Given Points

Q.

If the line passing through the points $\left(5,1,a\right)$ and $\left(3,b,1\right)$ crosses the $YZ$ plane at the point $\left(0,\frac{17}{2},\frac{-13}{2}\right)$, then:

1. $a=8,b=2$

2. $a=2,b=8$

3. $a=4,b=6$

4. $a=6,b=4$

Q. If a point $R(4,y,z)$ lies on the line segment joining the points $P(2,-3,4)$ and $Q(8,0,10)$, then the distance of $R$ from the origin is :

1. $\sqrt{53}$
2. $2\sqrt{14}$
3. $2\sqrt{21}$
4. $6$

Q. If a point $R(4,y,z)$ lies on the line segment joining the points $P(2,-3,4)$ and $Q(8,0,10)$, then the distance of $R$ from the origin is :

1. $\sqrt{53}$
2. $2\sqrt{14}$
3. $2\sqrt{21}$
4. $6$

Q. A line $L$ passing through origin is perpendicular to the lines
$L_1:\overrightarrow{r}=(3+t)\hat{i}+(-1+2t)\hat{j}+(4+2t)\hat{k}$
$L_2:\overrightarrow{r}=(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k}$
If the co-ordinates of the point in the first octant on $L_2$ at the distance of $\sqrt{17}$ from the point of intersection of $L$ and $L_1$ are $(a,b,c),$ then $18(a+b+c)$ is equal to

Q.

The line passing through the points $(5,1,a)\text{and}(3,b,1)$ crosses the $yz-$ plane at the point $(0,\frac{17}{2},\frac{-13}{2})$, then

1. $a=2,b=8$

2. $a=4,b=6$

3. $a=6,b=4$

4. $a=8,b=2$