formula. Example : Find the coordinates of the point which divides the line segment joining the points ( , – ) and ( , ) in the ratio : internally. Solution : Let P( x , y ) be the required point. Using the section formula, we get x = ( ) ( ) , y = ( ) (– ) Therefore, ( , ) is the required point.
Example : In what ratio does the point (– , ) divide the line segment joining the points A(– , ) and B( , – )? Solution : Let (– , ) divide AB internally in the ratio m : m . Using the section formula, we get (– , ) = – ( ) Recall that if ( x , y ) = ( a , b ) then x = a and y = b . – = and Now, – = gives us – m – m = m – m m = m m : m = : You should verify that the ratio satisfies the y -coordinate also.
Now, = (Dividing throughout by m ) = Therefore, the point (– , ) divides the line segment joining the points A(– , ) and B( , – ) in the ratio : . Alternatively : The ratio m : m can also be written as : , or k : . Let (– , ) divide AB internally in the ratio k : . Using the section formula, we get (– , ) = ( ) – = – k – = k – k = k : = : You can check for the y -coordinate also.
So, the point (– , ) divides the line segment joining the points A(– , ) and B( , – ) in the ratio : . Note : You can also find this ratio by calculating the distances PA