Question

If the point P (2,1) lies on the segment joining point A (4,2) and B (8,4) then
a. $AP=\dfrac{1}{3}AB$
b. $AB=PB$
c. $PB=\dfrac{1}{3}AB$
d. $AP=\dfrac{1}{2}AB$

Answer 1

Hint: In order to find the solution of this question, we will find the ratio of P dividing the line segment joining A and B as $k:1$. And then we will apply the section formulae, that is, if (x, y) divides the line joining $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ in the ratio m:n, then $x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}$ and $y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}$. And hence, we will get our answer.

Complete step-by-step solution -

In this question, we have been asked to find the ratio of the line segment joining P (2,1), A (4,2) and B (8,4).
To solve this question, we will first consider the ratio of P (2,1) dividing the line joining A (4,2) and B (8,4) as k:1. And we know that if (x, y) divides the line joining $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ in the ratio m:n, then $x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}$ and $y=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}$. So, we can say that if P (2,1) denotes (x, y), A (4,2) denotes $\left( {{x}_{1}},{{y}_{1}} \right)$, B (8,4) denotes $\left( {{x}_{2}},{{y}_{2}} \right)$ and k:1 denotes the ratio m:n, then by using the section formula, that is, $x=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}$ and on substituting the value in it, we will get,
$2=\dfrac{k\left( 8 \right)+1\left( 4 \right)}{k+1}$
Now, we will simplify it further. So, we get,
2k + 2 = 8k + 4
Taking the terms containing the variable k on one side and the rest all terms on the other side, we get,
8k – 2k = 2 – 4
6k = -2
$k=-\dfrac{2}{6}=-\dfrac{1}{3}$
Hence, we get the ratio of P (2,1) dividing the line joining A (4,2) and B (8,4) as $-\dfrac{1}{3}$, that is it divides the line externally in the ratio 1:3. We can represent it as follows
Hence, we can say that if PA = x, then PB will be 3x. So, we can say that AB = PB – PA = 3x – x = 2x. Hence, we can say,
$\begin{align}
  & \dfrac{PA}{AB}=\dfrac{x}{2x} \\
 & \dfrac{PA}{AB}=\dfrac{1}{2} \\
 & PA=\dfrac{1}{2}AB \\
\end{align}$
Hence, we obtained the relation between the line segments joining P, A and B as $AP=\dfrac{1}{2}AB$. Therefore, option (d) is the correct answer.

Note: While solving this question, we can also use the relation of the y coordinates of the points P, A and B to get the ratio k:1. Also, we can verify our answer by finding the value of k from both the coordinates. Also, the possible mistake one can make in this question is by choosing the option (a), that is $AP=\dfrac{1}{3}AB$, but this is wrong, as we have obtained $\dfrac{PA}{PB}=\dfrac{1}{3}$ and not $\dfrac{PA}{AB}=\dfrac{1}{3}$.