First slide

Straight lines in 3D

Question

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

Moderate

A

(-3,2, 1)
B

(1,2,2)
C

(4,5,3)
D

none of these

Solution

Let AD be the perpendicular and D be the foot of the perpendicular which divides .BC in the ratio $λ : 1$ ,
then $D (\frac{10 λ - 9}{λ + 1}, \frac{4}{λ + 1}, \frac{- λ + 5}{λ + 1})$
The direction ratios of AD, $\frac{10 λ - 9}{λ + 1}, \frac{4}{λ + 1}$ and $\frac{- λ + 5}{λ + 1}$
and direction ratios of BC are 19, - 4 and -6.
$Since A D ⊥ B C, we get$
$\begin{array}{l} 19 (\frac{10 λ - 9}{λ + 1}) - 4 (\frac{4}{λ + 1}) - 6 (\frac{- λ + 5}{λ + 1}) = 0 \\ or λ = \frac{31}{28} \end{array}$
Hence, on putting the value of $λ$ in (i), we get required foot of the perpendicular, i.e., $(\frac{58}{59}, \frac{112}{59}, \frac{109}{59})$

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