Question

find the ratio in which the point (X, 2) divides the line segment joining the points of A(12,5) and B(4,-3).also find the value of X.

Answer 1

Since P divides the line segment joining points A(12,5) and B(4,-3) therefore,
x= \frac{m x_{2}+n x_{1} }{m+n}x=​m+n​​mx​2​​+nx​1​​​​ and
y= \frac{m y_{2}+n y_{1} }{m+n}y=​m+n​​my​2​​+ny​1​​​​ 
where m:n is the ratio;
(x₁,y₁)=(12,5) and (x₂,y₂)=(4,-3) and (x,y)=(x,2) (coordinate of P)
∴, 2= \frac{-3m+5n}{m+n}2=​m+n​​−3m+5n​​ 
or, 2m+2n=-3m+5n2m+2n=−3m+5n 
or, 2m+3m=5n-2n2m+3m=5n−2n 
or, 5m=3n5m=3n 
or, \frac{m}{n}= \frac{3}{5}​n​​m​​=​5​​3​​ 
∴, x= \frac{3.4+5.12}{3+5}x=​3+5​​3.4+5.12​​ 
or, x= \frac{12+60}{8}x=​8​​12+60​​ 
or, x=72/8x=72/8 
or, x=9x=9 
∴, The required ratio is \frac{3}{5}​5​​3​​ and x=9x=9 . Ans.

Answer 2

Answer:

Let the required ratio be K:1.

Then, By section formula,the Coordinates of P are :

P ( 4K + 12/K + 1 , -3K + 5 / K + 1 )

But , this points is given as P ( x , 2).

Therefore,

⇒ -3K + 5 / K + 1 = 2

⇒ -3K + 5 = 2K + 2

⇒ 5K = 3

⇒ K = 3/5.

So, the required ratio is 3:5.