Question

A (8,3),b(x,y),c(8,4) divides ab in the ratio 1:5 internally then b=

Answer 1

Given : c(8,4) divides ab in the ratio 1:5  A ( 8 , 3) , B = ( x , y)

To find : Cordinate of B

Solution:

c(8,4) divides ab in the ratio 1:5

A ( 8 , 3)

B = ( x , y)

Ratio 1  :  5

interal dividing formula in m : n ratio

(mx₂ + nx₁)/(m+n)  , (my₂ + ny₁)/(m+n)  

=> 8  =  (1 * x  + 5 * 8)/(1 + 5)    , 4  =  ( 1*y  + 5 * 3)/(1 + 5)

=> 48 = x + 40     ,  24  = y + 15

=> x = 8    ,  y = 9

=> B = ( 8 , 9)

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Answer 2

Given:

AB is a line segment with coordinates A(8, 3) and B(x, y)

Point C(8, 4) divides AB internally in the ratio 1:5

To find:

The coordinates of B

Solution:

To solve the above given question we will use the section formula.

The section formula gives the coordinates of the point P(x, y) which divides a line segment with coordinates A(x₁, y₁) & B(x₂, y₂) internally in the ratio of m:n as

$\boxed{\bold{P(x, y)\:=\:[ \:\frac{mx_2\:+\:nx_1}{m\:+\:n} \:,\: \frac{my_2\:+\:ny_1}{m\:+\:n} \:]}}$

Here we have

P(x, y) = C(8, 4)

m = 1

n = 5

A(x₁, y₁) = A(8, 3)

B(x₂, y₂) = B(x, y)

Now, substituting all the given values in the section formula, we get

$C(8, 4)\:=\:[ \:\frac{(1\times x)\:+\:(5\times8)}{1\:+\:5} \:,\: \frac{(1\times y)\:+\:(5\times 3)}{1\:+\:5} \:]$

⇒ $C(8, 4)\:=\:[ \:\frac{ x\:+\:40}{6} \:,\: \frac{ y\:+\:15}{6} \:]$

∴ $8\:=\:\frac{ x\:+\:40}{6}$

⇒ $48 = x\:+\:40$

⇒ $x\:=\:48 \:-\:40$

⇒ $\bold{x = 8}$

and

∴ $4\:=\:\frac{ y\:+\:15}{6}$

⇒ $24 = y\:+\:15$

⇒ $y\:=\:24 \:-\:15$

⇒ $\bold{y = 9}$

Thus, B(x, y) = B(8, 9).

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