Question

Provide the answer to this question. P(p, 5) lies on the line joining A(1, 2) and B(-4, 7). Find the ratio dividing the line internally AP:PB *

Answer 1

Answer :

The required ratio is 3 : 2

Given :

• The points are A(1 , 2) and B(-4 ,7)
• P(p , 5) is another lies on the line joining A and B

To Find :

• The ratio in which point P divides the line AB

Formula to be used :

• If (x₁, y₁) and (x₂,y₂) are two points and (x , y) divides the line in ratio m:n then the coordinates of x and y are given by :
• $\sf{x=\dfrac{mx_2 + nx_1}{m+n}}$

• $\sf{y = \dfrac{my_2 + ny_{1}}{m+n}}$

Solution :

Let us consider the ratio in which P point divides AB be m:n

According to section formula :

$\implies \sf{p= \dfrac{-4m+n}{m+n}}$

and for Y-coordinate

$\sf {\implies5= \dfrac{7m + 2n}{m+n}}\\\\ \sf{\implies5(m+n)=7m + 2n}\\\\ \sf{\implies5m + 5n = 7m +2n}\\\\ \sf{\implies 5n - 2n = 7m - 5m}\\\\ \sf{\implies 3n = 2m }\\\\ \sf{\implies\dfrac{3}{2}=\dfrac{m}{n}}\\\\ \sf{\implies\dfrac{m}{n}=\dfrac{3}{2}}\\\\ \sf{\implies m:n = 3:2}$

or AP : BP = 3:2

Answer 2

Step-by-step explanation:

$\bf \huge \: Question \:$

• Provide the answer to this question. P(p, 5) lies on the line joining A(1, 2) and B(-4, 7). Find the ratio dividing the line internally AP:PB *

____________________________

$\bf \huge \: To \:Find$

• the ratio dividing the line internally AP:PB
• Provide the answer to this question. P(p, 5) lies on the line joining A(1, 2) and B(-4, 7)

_____________________________

$\bf \huge \: Given \:$

p(p, 5) lies on the line joining A(1, 2) and B(-4, 7).

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$\bf \: Aplying \: The\: Formula$

$\bf \: \:First\: Formula$

$\bf{x=\dfrac{mx_2 + nx_1}{m+n}}$

$\bf \: \:secomd\: Formula$

$\bf{y = \dfrac{my_2 + ny_2}{m+n}}$

______________________________

Given consider the ratio in which P point divides AB be m:n

According to the formula :

Putting the value :-

$\implies \bf{p= \dfrac{-4m+n}{m+n}}$

______________________________

GIVEN in the Question

Y-coordinate

$\bf\: \implies5= \dfrac{7m + 2n}{m+n}\\\\ \bf\:\implies5(m+n)=7m + 2n\\\\ \bf\:\implies5m + 5n = 7m +2n\\\\ \bf\:\implies 5n - 2n = 7m - 5m\\\\ \bf\: on \:substraction\\\\\:\bf\:\implies 3n = 2m \\\\\\\\\:\bf\:On\: bringing\: 3 \:from \:one \:side\: to \:another\\\\\\\\\</p><p> \bf\:\implies\dfrac{3}{2}=\dfrac{m}{n}\\\\ \bf\:\implies\dfrac{m}{n}=\dfrac{3}{2}\\\\ \bf\:Y-coordinates m:n = 3:2$

$\bf\:\red {AP : BP = 3:2}$

The ratio dividing the line internally $\bf</u></strong><strong><u>\:</u></strong><strong><u>\red {AP : BP = 3:2}$