If Px- y- z is a point on the line segment joining Q2- 3- 4 and R3- 5- 6 such that the projections of the vector OP on the coordinate axes are 135-215-265 respectively- where O denotes the origin- then P divides QR in the ratio

Let P(x,y,z) divide QR in the ratio of k:1 Any point P is (3k+2k+1,5k+3k+1,6k+4k+1) Since the projection of the vector OP on the x axis is 135, ∴3k+2k+1=135 ⇒ ...

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Standard X

Mathematics

Trigonometric Ratios

If Px, y, z i...

Question

If $P (x, y, z)$ is a point on the line segment joining $Q (2, 3, 4)$ and $R (3, 5, 6)$ such that the projections of the vector $- - \to O P$ on the coordinate axes are $\frac{13}{5}, \frac{21}{5}, \frac{26}{5}$ respectively, where $O$ denotes the origin, then $P$ divides $Q R$ in the ratio

2 : 3

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3 : 1

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1 : 3

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3 : 2

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Solution

The correct option is D $3 : 2$
Let $P (x, y, z)$ divide $Q R$ in the ratio of $k : 1$

Any point $P$ is $(\frac{3 k + 2}{k + 1}, \frac{5 k + 3}{k + 1}, \frac{6 k + 4}{k + 1})$
Since the projection of the vector $- - \to O P$ on the $x -$ axis is $\frac{13}{5},$
$∴ \frac{3 k + 2}{k + 1} = \frac{13}{5}$
$\Rightarrow 15 k + 10 = 13 k + 13$
$\Rightarrow k = \frac{3}{2}$
Hence, the ratio is $\frac{3}{2} : 1$ or $3 : 2$

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If P(x,y,z) is a point on the line segment joining Q(2,3,4) and R(3,5,6) such that the projections of the vector −−→OP on the coordinate axes are 135,215,265 respectively, where O denotes the origin, then P divides QR in the ratio

The correct option is D 3:2Let P(x,y,z) divide QR in the ratio of k:1 Any point P is (3k+2k+1,5k+3k+1,6k+4k+1) Since the projection of the vector −−→OP on the x−axis is 135, ∴3k+2k+1=135 ⇒15k+10=13k+13 ⇒k=32 Hence, the ratio is 32:1 or 3:2

If $P (x, y, z)$ is a point on the line segment joining $Q (2, 3, 4)$ and $R (3, 5, 6)$ such that the projections of the vector $- - \to O P$ on the coordinate axes are $\frac{13}{5}, \frac{21}{5}, \frac{26}{5}$ respectively, where $O$ denotes the origin, then $P$ divides $Q R$ in the ratio