Question

Find the ratio in which $ P(4,m) $ divides the line segment joining the points $ A(2,3) $ and $ B(6, - 3). $ Hence find m.

Answer 1

Hint: Here we will use the section formula for the given points and will assume that the ratio be $ m:n $ and then will simplify the mathematical expression for the required resultant value. While simplification assume the ratio to be $ k:1 $

Complete step-by-step answer:
Apply the section formula for the points in which $ P(4,m) $ divides the line segment joining the points $ A(2,3) $ and $ B(6, - 3). $ in the ratio of $ m:n. $
Then the expression becomes –
 $ (x,y) = \left( {\dfrac{{m{x_1} + n{x_2}}}{{m + n}},\dfrac{{m{y_1} + n{y_2}}}{{m + n}}} \right) $
Let us assume that P divides AB in the ratio of $ k:1 $
 $ (4,m) = \left( {\dfrac{{2k + 1(6)}}{{k + 1}},\dfrac{{3k + 1( - 3)}}{{k + 1}}} \right) $
Simplify the above expression –
 $ (4,m) = \left( {\dfrac{{2k + 6}}{{k + 1}},\dfrac{{3k - 3}}{{k + 1}}} \right) $
Compare both the sides of the equation –
 $ 4 = \dfrac{{2k + 6}}{{k + 1}} $ … (A) and $ m = \dfrac{{3k - 3}}{{k + 1}} $ … (B)
Cross multiply the above expression, where the denominator of one side is multiplied with the numerator of the opposite side.
 $ 4(k + 1) = 2k + 6 $
Multiply the term inside the bracket with the terms inside the bracket.
 $ 4k + 4 = 2k + 6 $
Move all the constants on one side and the term with the variable on one side. When you move any term from one side of the equation to the opposite side then the sign of the term changes. Positive term changes to negative and vice-versa.
 $ 4k - 2k = 6 - 4 $
Simplify the expression finding the difference of the terms.
 $ 2k = 2 $
Term multiplicative on one side if moved to the opposite then it goes to the denominator.
 $
  k = \dfrac{2}{2} \\
  k = 1 \;
  $
Place above value in the equation (B)
 $ m = \dfrac{{3(1) - 3}}{{1 + 1}} $
Simplify the above equation –
 $ m = \dfrac{{3 - 3}}{2} $
Find the difference of the term on the numerator –
 $ m = \dfrac{0}{2} $
 $ m = 0 $
This is the required solution.

Note: Always remember that like terms with the same value and opposite sign cancels each other. Also, zero upon any number is always zero. Be careful while comparing the coordinates of the points, x coordinate is equal to the x coordinate of the point on the opposite side of the equation and the same is applied for y coordinate.