Question

The co-ordinates of the foot of perpendicular drawn from the origin to the line joining the points (–9, 4, 5) and (10, 0, –1) will be

• A. (– 3, 2, 1)
• B. (1, 2, 2)
• C. (4, 5, 3)
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Let AD be perpendicular and D be foot of perpendicular which divide BC in ratio then

$D \left( \frac { 10 \lambda - 9 } { \lambda + 1 } , \frac { 4 } { \lambda + 1 } , \frac { - \lambda + 5 } { \lambda + 1 } \right)$ …..(i)

The direction ratio of AD are $\frac { 10 \lambda - 9 } { \lambda + 1 } , \frac { 4 } { \lambda + 1 } , \frac { - \lambda + 5 } { \lambda + 1 }$ and direction ratio of BC are 19, – 4 and – 6.

Since $A D \perp B C$

$\Rightarrow 19 \left( \frac { 10 \lambda - 9 } { \lambda + 1 } \right) - 4 \left( \frac { 4 } { \lambda + 1 } \right) - 6 \left( \frac { - \lambda + 5 } { \lambda + 1 } \right) = 0$

$\Rightarrow \lambda = \frac { 31 } { 28 }$ .

Hence on putting the value of $\lambda$ in (i), we get required foot of the perpendicular i.e., $\left( \frac { 58 } { 59 } , \frac { 112 } { 59 } , \frac { 109 } { 59 } \right)$.

Trick: The line passing through these points is $\frac { x + 9 } { 19 } = \frac { y - 4 } { - 4 } = \frac { z - 5 } { - 6 }$ Now co-ordinates of the foot lie on this line, so they must satisfy the given line. But here no point satisfies the line, hence answer is (4).