Question

6.
Consider the following figure.
(i)
Find the distance PO.
() Find the co-ordinates of the point which divides the line segment ini
points P and Q internally in the ratio 2:3.​

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 PO

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

Using the formula for distance between two points, we get

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

$\quad=\sqrt{3^{2}+4^{2}+5^{2}}$ units

$\quad=\sqrt{9+16+25}$ units

$\quad=\sqrt{50}$ units

$\quad=\bold{5\sqrt{2}}$ units

(ii) 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

(iii) 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., $\bold{\big(\frac{9}{5},\:4,\:2\big)}.$

Distance related problems:

Q1. Find the distance between mid-point of line segment AB with A (2, - 7), B (9, - 2) and the point C (5, - 6).

Go here: https://brainly.in/question/11839327

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

Go here: https://brainly.in/question/2367146

Answer 2

Consider the following figure.

(i) the distance PO=5√2 units.

(ii) the co-ordinates of the point which divides the line segment in points P and Q internally in the ratio 2:3=[(9/5), (4), (2)].

Stepwise explanation is given below:

- 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-

[(lx2+mx1)/(l+m),(ly2+my1)/(l+m),(lz2+mz1)/(l+m)]

- 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 PO

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

- Using the formula for distance between two points, we get

PO=√[(3-0)^2+(4-0)^2+(5-0)^2]

=√(3^2+4^2+5^2)units

=√(9+16+25) units

=√(50) units

=5√2 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

=[(2*0+3*3)/(2+3), (2*4+3*4)/(2+3), (2*5+3*0)/(2+3)]

=[(0+9)/5, (8+12)/5, (10+0)/5]

=[(9/5), (4), (2)]