Question

Let A(2, 3) and B(2, -4) be two points. Ir P lies on the x-axis, such that AP - ³/7 AB,
find the coordinates of P​

Answer 1

$\begin{gathered}\begin{gathered}\bf Given -  \begin{cases} &\sf{coordinates \: of \: A(2, 3)} \\ &\sf{coordinates \: of \: B(2, -4)}\\ &\sf{P \: lies \: on \: the \: x-axis}\\ &\sf{AP = \: \dfrac{3}{7} AB} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf  To \:  Find :-  \begin{cases} &\sf{the \: coordinates \: of \: P}  \end{cases}\end{gathered}\end{gathered}$

Concept Used

$\bold{ \red{Section \: Formula }}$

Let us consider a line segment joining the points A and B and let C divides AB in the ratio m : n internally, then coordinates of C(x, y) is given by

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

where coordinates of A and B are

$\sf \:  A(x_1,y_1) and B(x_2,y_2)$

$\large\underline\purple{\bold{ \tt{Solution} :- }}$

$\begin{gathered}\begin{gathered}\bf Let = \begin{cases} &\sf{the \: coordinates \: of \: P \: be \: (x, 0)} \\ \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\bf\red{According \: to \: statement}\end{gathered}$

$\tt \:  \longrightarrow \: AP = \dfrac{3}{7} \: AB \:$

$\tt \:  \longrightarrow \: 7AP = 3(AP + PB)$

$\tt \:  \longrightarrow \: 7AP = 3AP + 3PB$

$\tt \:  \longrightarrow \: 7AP - 3AP = 3PB$

$\tt \:  \longrightarrow \: 4AP = 3PB$

$\tt\implies \: \boxed{ \tt \:  \green{\dfrac{AP}{PB} = \dfrac{3}{4} }}$

$\bf\implies \: \boxed{ \blue{\bf \: P \: divides \: AB \: in \: the \: ratio \: 3 \: : \: 4 .}}$

☆ Using section Formula, P(x, 0) divides the line segment A(2, 3) and B(2, -4) in the ratio 3 : 4, we get

$\tt \:  \longrightarrow \: (x, 0) = \bigg(\dfrac{3 \times 2 + 4 \times 2}{3 + 4} , \dfrac{ - 4 \times 3 + 3 \times 4}{3 + 4} \bigg)$

$\tt \:  \longrightarrow \: (x, 0) = \bigg(\dfrac{14}{4} , \dfrac{0}{7} \bigg)$

$\tt \:  \longrightarrow \: (x, 0) = (2, 0)$

$\tt\implies \: \boxed{ \purple{x \: = \: 2}}$

$\tt\implies \: \large\boxed{ \tt \:  \red{coordinates \: of \: P \: is \: (2 \: , \: 0)}}$