Question

if (4.6) divides the distance between the paints (X,Y)
and (5,7) internally in the ratio 2:1. Finch
the value of ond x y.​

Answer 1

Answer:-

Given:

(4 , 6) divides the line segment joining the points (x , y) & (5 , 7) internally in the ratio 2 : 1.

using section formula,

i.e., The co - ordinates of a point which divides the line segment joining the points (x₁ , y₁) , (x₂ , y₂)

internally in the ratio m : n are :

$\sf \large \: (x \: ,\: y) = \bigg( \dfrac{mx_2 + nx_1}{m + n} \: \: , \: \: \dfrac{my_2 + ny_1}{m + n} \bigg)$

Let,

• x = 4

• y = 6

• x₁ = x

• y₁ = y

• x₂ = 5

• y₂ = 7

• m = 2

• n = 1

Hence,

$\implies \sf \: \: (4\: , \: 6) = \bigg( \dfrac{(2)( 5)+ (1)(x)}{2 + 1} \: \:, \: \: \dfrac{(2)(7) + (1)(y)}{2 + 1} \bigg) \\ \\ \implies \sf \: \: (4\: , \: 6) = \bigg( \dfrac{10+ x}{3} \: \: , \: \: \dfrac{y + 14}{3} \bigg) \\ \\ \implies \sf \:4 = \frac{10 + x}{3} \\ \\ \implies \sf \:12 = 10 + x \\ \\ \implies \sf \:12 - 10 = x \\ \\ \implies \boxed{\sf \:x = 2}$

Similarly,

$\: \implies \sf \:6 = \frac{14 + y}{3} \\ \\ \implies \sf \:18 = 14 + y \\ \\ \implies \sf \:18 - 14 = y \\ \\ \implies \boxed{\sf \:y = 4}$

∴ (x , y) = (2 , 4).

Answer 2

Answer:

$\huge\underbrace\mathfrak\color{lime}{ †\: Solution:–}$

Given:

(4 , 6) divides the line segment joining the points (x , y) & (5 , 7) internally in the ratio 2 : 1.

using section formula,

i.e., The co - ordinates of a point which divides the line segment joining the points (x₁ , y₁) , (x₂ , y₂)

internally in the ratio m : n are :

\sf \large \: (x \: ,\: y) = \bigg( \dfrac{mx_2 + nx_1}{m + n} \: \: , \: \: \dfrac{my_2 + ny_1}{m + n} \bigg)(x,y)=(

m+n

Let,

• x = 4
• y = 6
• x₁ = x
• y₁ = y
• x₂ = 5
• y₂ = 7
• m = 2
• n = 1

Hence,

$$\begin{lgathered}\implies \sf \: \: (4\: , \: 6) = \bigg( \dfrac{(2)( 5)+ (1)(x)}{2 + 1} \: \:, \: \: \dfrac{(2)(7) + (1)(y)}{2 + 1} \bigg) \\ \\ \implies \sf \: \: (4\: , \: 6) = \bigg( \dfrac{10+ x}{3} \: \: , \: \: \dfrac{y + 14}{3} \bigg) \\ \\ \implies \sf \:4 = \frac{10 + x}{3} \\ \\ \implies \sf \:12 = 10 + x \\ \\ \implies \sf \:12 - 10 = x \\ \\ \implies \boxed{\sf \:x = 2}\end{lgathered}$$

Similarly,

$$\begin{lgathered}\: \implies \sf \:6 = \frac{14 + y}{3} \\ \\ \implies \sf \:18 = 14 + y \\ \\ \implies \sf \:18 - 14 = y \\ \\ \implies \boxed{\sf \:y = 4}\end{lgathered}$$

∴ (x , y) = (2 , 4).