Question

Find the distance between the following pair of points (a+b,b+c) and (a-b,c-b)

Answer 1

The distance is $2 b \sqrt{2}$

Given:

Points “(a\quad +\quad b,\quad b\quad +\quad c)”and  

(“a\quad -\quad b”,“c\quad -\quad b”)

To find:

“Distance between” the two pair.

Answer:

A (a+b, b+c) and B (a-b, c-d)  

As per the given question  

$AB=\sqrt { ({ x }_{ 2 }-{ x }_{ 1 })^{ 2 }+({ y }_{ 2 }-{ y }_{ 1 })^{ 2 } }$

We know that,

${ x }_{ 1 }\quad =\quad a\quad +\quad b$

${ x }_{ 2 }\quad =\quad a\quad -\quad b$

${ y }_{ 1 }\quad =\quad b\quad +\quad c$

${ y }_{ 2 }\quad =\quad c\quad -\quad b$

$\therefore AB=\sqrt{(a-b-a-b)^{2}+(c-b-b-c)^{2}}$

$=\sqrt{(-2 b)^{2}+(-2 b)^{2}}$

$=\sqrt{4 b^{2}+4 b^{2}}$

$=\sqrt{8 b^{2}}$

$AB=2 b \sqrt{2}$

Answer 2

Step-by-step explanation:

Distance between points

(

x

1

,

y

1

)

and

(

x

2

,

y

2

)

=

√

(

x

2

−

x

1

)

2

+

(

y

2

−

y

1

)

2

Let's name the points as A(a + b, b + c) and B(a - b, c - b) .

AB =

√

(

x

2

−

x

1

)

2

+

(

y

2

−

y

1

)

2

⇒

A

B

2

=

(

a

+

b

−

(

a

−

b

)

)

2

+

(

b

+

c

−

(

c

−

b

)

)

2

⇒

A

B

2

=

4

b

2

+

4

b

2

=

8

b

2

⇒

A

B

=

2

√

2

b

Therefore, the distance between points (a + b, b + c) and (a – b, c – b) is

2

√

2

b

units.