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Q.
The coordinates of the foot of the perpendicular drawn from the origin to the line joining the points $(-9,4,5)$ and $(10,0,-1)$ will be
Three Dimensional Geometry
Solution:
Let $AD$ be the perpendicular and $D$ be the foot of the perpendicular which divides $B C$ in the ratio $\lambda: 1$, then
$D\left(\frac{10 \lambda-9}{\lambda+1}, \frac{4}{\lambda+1}, \frac{-\lambda+5}{\lambda+1}\right)\, \dots(i)$
The direction ratios of $AD$ are $\frac{10 \lambda-9}{\lambda+1}, \frac{4}{\lambda+1}$ and $\frac{-\lambda+5}{\lambda+1}$
and direction ratios of BC are 1 9 ,-4 and - 6
Since $AD \perp BC$, we get
$19\left(\frac{10 \lambda-9}{\lambda+1}\right)-4\left(\frac{4}{\lambda+1}\right)-6\left(\frac{-\lambda+5}{\lambda+1}\right)=0$
or $\lambda=\frac{31}{28}$
Hence, on putting the value of $\lambda$ in (i), we get required foot of the perpendicular, i.e., $\left(\frac{58}{59}, \frac{112}{59}, \frac{109}{59}\right)$
