Question

16. Show that the points O(0, 0, 0), A(2,–3, 3), B(–2, 3, –3) are collinear. Find the ratio in which each point divides the segment joining the other two.

Answer 1

Answer:

Let's find area of △OAB which is given by,

Δ=

2

1

​

∣OA∣∣OB∣sinθ

where θ is angle between  

OA

 and  

OB

Now, cosθ=

∣

∣

∣

∣

​

 

OA

 

∣

∣

∣

∣

​

 

∣

∣

∣

∣

​

 

OB

 

∣

∣

∣

∣

​

OA

.

OA

​

=

4+9+9

​

 

4+9+9

​

−4−9−9

​

=

22

−22

​

⇒ cosθ=−1⇒ θ=180

o

⇒ Δ=0 as sin180=0

⇒ O,A,B are collinear

Now, let A divides  

OB

 in k:1

⇒ 2=

k+1

−2k+0

​

⇒ 2k+2=−2k⇒ 4k=−2⇒ k=−

2

1

​

⇒ A divides  

OB

 in 1:2 externally

Now for B

⇒ −2=

k+1

2k+0

​

⇒ −2k−2=2k⇒ 4k=−2

⇒ k=−

2

1

​

⇒ B divides  

OA

 in 1:2 externally for O

⇒ O=

k+1

−2k+2

​

⇒ 2k=2⇒ k=1

⇒ O divides  

AB

 in 1:1 internally

Step-by-step explanation: