Internal Division an interval in a given ratio is that if the interval \( (x_1,y_1) \) and \( (x_2,y_2) \) is divided in the ratio \( m:n \) then the coordinates are;
$$ \large \Big(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\Big)$$
Example 1: Basics
If \(A\) and \(B\) are the points \( (-4,3) \) and \( (2,-1) \) respectively, find the coordinates of \(P\) such that \( AP:PB=3:1\).
\( \begin{aligned} \displaystyle
P(x,y) &= \Big(\frac{3 \times 2 + 1 \times -4}{3+1},\frac{3 \times -1 + 1 \times 3}{3+1}\Big) \\
&= \Big(\frac{1}{2},0\Big) \\
\end{aligned} \\ \)
Example 2: Negative Ratio of Internal Division of Line Segments
If \(A\) and \(B\) are the points \( (-4,3) \) and \( (2,-1) \) respectively, find the coordinates of \(P\) such that \( AP:PB=-4:5\).
\( \begin{aligned} \displaystyle
P(x,y) &= \Big(\frac{-4 \times 2 + 5 \times -4}{-4+5},\frac{-4 \times -1 + 5 \times 3}{-4+5}\Big) \\
&= (-28,19) \\
\end{aligned} \\ \)
Example 3: Three Equal Parts using Internal Division of Line Segments
Divide the interval between \( (-1,1) \) and \( (5,10) \) into three equal parts.
\( \begin{aligned} \displaystyle
(a,b) &= \Big(\frac{1 \times 5 + 2 \times -1}{1+2},\frac{1 \times 10 + 2 \times 1}{1+2}\Big) \\
&= (1,4) \\
(c,d) &= \Big(\frac{2 \times 5 + 1 \times -1}{2+1},\frac{2 \times 10 + 1 \times 1}{2+1}\Big) \\
&= (3,7) \\
\end{aligned} \\ \)
Example 4: Finding the Ratio of Internal Division of Line Segments
If the point \( (-3,8) \) divides the interval between \( (6,-4) \) and \( (0,4) \) internally in the ratio \( k:1 \), find the value of \(k\).
\( \begin{aligned} \displaystyle
(-3,8) &= \Big(\frac{k \times 0 + 1 \times 6}{k+1},\frac{k \times 4 + 1 \times -4}{k+1}\Big) \\
&= \Big(\frac{6}{k+1},\frac{4k-4}{k+1}\Big) \\
\frac{6}{k+1} &= -3 \\
k+1 &= 6 \div -3 \\
k+1 &= -2 \\
\therefore k &= -3 \\
\end{aligned} \\ \)
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