Question

Find the distance PQ
1.Find the co ordibates of the point 2.which divides the line segment joining the points PandQ in ratio2:3

Answer 1

Let us know some formulae before we proceed to solve the problem:

1. If P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) be any the co-ordinates of any two points, the distance between them is given by

PQ = $\mathsf{\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}}.}$

2. If the join of two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) is divided internally by another point R into the ratio l : m, the co-ordinates of R be

$\big(\frac{lx_{2}+mx_{1}}{l+m},\:\frac{ly_{2}+my_{1}}{l+m},\:\frac{lz_{2}+mz_{1}}{l+m}\big)$

Now we solve the given problem:

First we write the co-ordinates of the points O, P and Q, which are (0, 0, 0), (3, 4, 5) and (0, 4, 0) respectively.

(i) To find the distance PQ

The given points are P (3, 4, 5) and Q (0, 4, 0).

Using the formula for distance between two points, we get

PQ $\mathsf{=\sqrt{(3-0)^{2}+(4-4)^{2}+(5-0)^{2}}}$ units

$\quad=\sqrt{3^{2}+0^{2}+5^{2}}$ units

$\quad=\sqrt{9+0+25}$ units

$\quad=\bold{\sqrt{34}}$ units

(ii) To find the co-ordinates of the desired point

The join of the points P (3, 4, 5) and Q (0, 4, 0) are divided internally into the ratio 2 : 3 by another point, whose co-ordinates be

$\quad\big(\frac{2\times 0+3\times 3}{2+3},\:\frac{2\times 4+3\times 4}{2+3},\:\frac{2\times 5+3\times 0}{2+3}\big)$

i.e., $\big(\frac{0+9}{5},\:\frac{8+12}{5},\:\frac{10+0}{5}\big)$

i.e., $\bold{\big(\frac{14}{5},\:4,\:2\big)}.$

Distance related problems:

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Q2. Problem from co-ordinates geometry (3d).

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Q3. Similar problem can be seen from here.

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