Question

If P(x, y, z) is a point on the line segment joining Q(2, 2, 4) and R (3, 5, 6) such that the projections of $\overset{\rightarrow}{OP}$on the axes are $\frac{13}{5}$, $\frac{19}{5}$, $\frac{26}{5}$ respectively, then P divides QR in the ratio –

• A. 1 : 2
• B. 3 : 2
• C. 2 : 3
• D. 1 :

Answer 1

Answer: (A)

1 : 2

Answer 2

Since$\overset{\rightarrow}{OP}$ has projections $\frac{13}{5}$, $\frac{19}{5}$and $\frac{26}{5}$ on the

co-ordinate axes, therefore $\overset{\rightarrow}{OP}$= $\frac { 19 } { 5 }$ $\widehat{j}$+$\frac { 26 } { 5 }$ $\widehat{k}$.

Suppose P divides the join of Q (2, 2, 4) and R (3, 5, 6) in the ratio l : 1. Then the position vector of P is

$\left( \frac{3\lambda + 2}{\lambda + 1} \right)\widehat{i}$+$\left( \frac{5\lambda + 2}{\lambda + 1} \right)\widehat{j}$ +$\left( \frac{6\lambda + 4}{\lambda + 1} \right)\widehat{k}$

\$\frac { 19 } { 5 }$ $\widehat{j}$+$\frac { 26 } { 5 }$ $\widehat{k}$

= $\left( \frac{3\lambda + 2}{\lambda + 1} \right)\widehat{i}$+$\left( \frac{5\lambda + 2}{\lambda + 1} \right)\widehat{j}$ +$\left( \frac{6\lambda + 4}{\lambda + 1} \right)\widehat{k}$

Ž $\frac{3\lambda + 2}{\lambda + 1}$ = $\frac{13}{5}$, $\frac{5\lambda + 2}{\lambda + 1}$ = $\frac{19}{5}$, $\frac{6\lambda + 4}{\lambda + 1}$ = $\frac{26}{5}$ Ž l = $\frac{3}{2}$.