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.

Hint: For solving this problem, we first assume the ratio in which the line segment is divided in variables. Now, by using the section formula between two points, we evaluate the value of those variables. This gives us the ratio in which the line segment is divided. Finally, we can obtain the value of the required quantity.

Complete Step-by-Step solution:
According to our question, we are given that point P (4, m) divides the line segment joining the points A (2, 3) and B (6, -3). Now, let the ratio in which the point P divides line segment AB be 1: k. We know that the section formula states that if a point P (x, y) lies on line segment AB joining the points A $\left( {{x}_{1}},{{y}_{1}} \right)$ and B $\left( {{x}_{2}},{{y}_{2}} \right)$ and satisfies the relation $AP:PB=1:k$. Then, the coordinates of point of division can be expressed as:
$P=\left( \dfrac{{{x}_{2}}+k{{x}_{1}}}{1+k},\dfrac{{{y}_{2}}+k{{y}_{1}}}{1+k} \right)$
Now, putting the values of coordinates of P as (4, m), A (2, 3) and B (6, -3) in the respective places, we get
$\left( 4,m \right)=\left( \dfrac{6+2k}{1+k},\dfrac{-3+3k}{1+k} \right)$
Now, to evaluate the value of k, we equate the x coordinates:
\begin{align} & 4=\dfrac{6+2k}{1+k} \\ & 4(1+k)=6+2k \\ & 4+4k=6+2k \\ & 2k=2 \\ & k=1 \\ \end{align}
Hence, the value of k is 1 and the ratio will be 1: 1.
Now, to evaluate m, we equate the y coordinates:
\begin{align} & m=\dfrac{-3+3k}{1+k} \\ & m=\dfrac{-3+3}{2} \\ & m=0 \\ \end{align}
Therefore, the value of m is equal to 0.
So, the coordinates of P can be expressed as (4, 0).

Note: The key concept for solving this problem is the knowledge of section formula and the assumption of ratio when two coordinates of line segment are given. This knowledge is helpful in solving complex problems.