## Internal Division of Line Segments

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;

$$\Big(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\Big)$$

### Basic 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 \( AP:PB=3:1\).

### 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 \( AP:PB=-4:5\).

### Three Equal Parts using Internal Division of Line Segments

Divide the interval between \( (-1,1) \) and \( (5,10) \) into three equal parts.

### 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 \(k\).