Search any question & find its solution
Question:
Answered & Verified by Expert
A point $R$ with $x$-coordinate 4 lies on the line segment joining the points $P(2,-3,4)$ and $Q(8,0,10)$. Find the coordinates of the point $R$.
Solution:
2442 Upvotes
Verified Answer
Let $R$ divides $P Q$ in the ratio $k: 1$ where $P$ is $(2,-3,4)$ and $Q$ is $(8,0,10)$
$\therefore \quad$ The coordinate of point $R$ are
$\left(\frac{8 k+2}{k+1}, \frac{0-3}{k+1}, \frac{10 k+4}{k+1}\right)$
But $x$-coordinate of $R$ is $4, \frac{8 k+2}{k+1}=4$
or $8 k+2=4 k+4$ or $4 k=2$ or $k=\frac{1}{2}$
$y_1=\frac{-3}{k+1}=\frac{-3}{\frac{1}{2}+1}=-2 \text {, }$
$z_1=\frac{10 k+4}{k+1}=\frac{10 \times \frac{1}{2}+4}{\frac{1}{2}+1}=\frac{5+4}{3 / 2}=9 \times \frac{2}{3}=6$
$\therefore$ The point $R$ is $(4,-2,6)$
$\therefore \quad$ The coordinate of point $R$ are
$\left(\frac{8 k+2}{k+1}, \frac{0-3}{k+1}, \frac{10 k+4}{k+1}\right)$
But $x$-coordinate of $R$ is $4, \frac{8 k+2}{k+1}=4$
or $8 k+2=4 k+4$ or $4 k=2$ or $k=\frac{1}{2}$
$y_1=\frac{-3}{k+1}=\frac{-3}{\frac{1}{2}+1}=-2 \text {, }$
$z_1=\frac{10 k+4}{k+1}=\frac{10 \times \frac{1}{2}+4}{\frac{1}{2}+1}=\frac{5+4}{3 / 2}=9 \times \frac{2}{3}=6$
$\therefore$ The point $R$ is $(4,-2,6)$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.