Question

if and b are(-2,-2)and(2,-4)respectively. find the coordinate of p such that Ap=7/3AB and p lies on the line segment AB​

Answer 1

Solution:-

$\setlength{\unitlength}{2.5 cm}\begin{picture}(2,2)\thicklines\put(2,4){\line(2,0){3}}\put(3.5,4.1){\line(0,-1){0.2}}\put(1.8,3.8){\sf A(2,-2)}\put(3.5,3.7){\sf p}\put(4.8,3.8){\sf B (2,-4)}\put(2.7,4.1){\sf 3}\put(4,4.1){\sf 4}\end{picture}$

:- Given co -ordinate is

=> A( - 2 , - 2 )  , B( 2 , - 4 )

:- Ratio are

$\sf \to \:AP = \dfrac{3}{7} AB$

$\sf \to7AP = 3AB$

$\sf \to \:7AP = 3(AP + BP)$

$\sf \to \: 7AP = 3AP + 3BP$

$\sf \to \: 7AP - 3AP = 3BP$

$\sf \to \: 4AP = 3BP$

$\to \sf \: \dfrac{AP}{BP} = \dfrac{3}{4}$

:- Given ratio is 3 : 4

:- Formula

$\sf \: x = \dfrac{mx_{2} + nx _{1}}{m + n} \: and \: \sf \: y= \dfrac{my_{2} + ny _{1}}{m + n} \:$

So

$\sf \: \to \: m = 3 \: ,n \: = 4,x _{1} = - 2 ,x _{2} = - 2, y _{1} = 2, y _{2} = - 4$

$\sf \to \: x = \dfrac{3 \times - 2 + 4 \times - 2}{3 + 4} \: \: and \: \: y = \dfrac{3 \times 2 + 4 \times - 4}{3 + 4}$

$\sf\to \: x = \dfrac{ - 6 - 8}{7} \: \: and \: \: y = \dfrac{6 - 8}{7}$

$\sf \: \to \: x = \dfrac{ - 14}{7} \: and \: y = \dfrac{ - 2}{7}$