Straight Line and Pair of Straight Lines 1 Question 7
7. 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
(2019 Main, 8 April II)
(a) $2 \sqrt{21}$
(b) $\sqrt{53}$
(c) $2 \sqrt{14}$
(d) 6
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Solution:
- Given points are $P(2,-3,4), Q(8,0,10)$ and $R(4, y, z)$. Now, equation of line passing through points $P$ and $Q$ is $\frac{x-8}{6}=\frac{y-0}{3}=\frac{z-10}{6}$
[Since equation of a line passing through two points $A\left(x _1, y _1, z _1\right)$ and $B\left(x _2, y _2, z _2\right)$ is given by
$$ \begin{aligned} & \frac{x-x _1}{x _2-x _1}=\frac{y-y _1}{y _2-y _1}=\frac{z-z _1}{z _2-z _1} \\ & \Rightarrow \quad \frac{x-8}{2}=\frac{y}{1}=\frac{z-10}{2} \end{aligned} $$
$\because \quad$ Points $P, Q$ and $R$ are collinear, so
$$ \begin{aligned} & \frac{4-8}{2} & =\frac{y}{1} & =\frac{z-10}{2} \\ \Rightarrow & -2 & =y & =\frac{z-10}{2} \\ \Rightarrow & & y & =-2 \\ & \text { and } & & =6 \end{aligned} $$
So, point $R$ is $(4,-2,6)$, therefore the distance of point $R$ from origin is
$$ \begin{aligned} & \quad O R=\sqrt{16+4+36} \\ & =\sqrt{56}=2 \sqrt{14} \end{aligned} $$
