Question

Find the ratio in which the point $$P(3,5)$$ divides $\overline{AB}$ where $$A(1,3)$$ and $B(7,9)$.

Answer 1

Hint: We will first start defining the section formula and then we will apply the section formula for internal division that is $P(a,b)=({\displaystyle \frac{x{a}_2+y{a}_1}{x+y}},{\displaystyle \frac{x{b}_2+y{b}_1}{x+y}})$ to our given points and then get our ratio and hence, the answer. We have the points as $P(a,b)=P(3,5)$ , $A(a_1,b_1)=A(1,3)$ and $B(a_2,b_2)=B(7,9)$ .

Complete step by step answer:

To find the ratio in which the point $$P(3,5)$$ divides $\overline{AB}$ where $$A(1,3)$$ and $B(7,9)$ , we will use the section formula. First let’s define what a section formula is.

So basically, the section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio $x:y$ as shown in the following figure:

Now, let’s see the formula of the internal division with section formula:

If point $P(a,b)$ lies on line segment $AB$ (between points $A$ and $B$) and satisfies $$AP:PB=x:y$$ , then we say that $P$ divides $$AB$$ internally in the ratio $x:y$ . The point of division has the coordinates:
$P(a,b)=({\displaystyle \frac{x{a}_2+y{a}_1}{x+y}},{\displaystyle \frac{x{b}_2+y{b}_1}{x+y}})$
So, let’s take the question, and see it is given that the point $$P(3,5)$$ divides $\overline{AB}$ where $$A(1,3)$$ and $B(7,9)$ :
Now, we can solve this question in normal method by assuming the ratio to be $x:y$ . But now we will take a different substitution. Let’s assume that $k={\displaystyle \frac{x}{y}}$ so that $x:y=k:1$ .
Now the required ratio will be $k:1$, now the formula for internal division will become:
$$P(a,b)=({\displaystyle \frac{k{a}_2+a_1}{k+1}},{\displaystyle \frac{k{b}_2+b_1}{k+1}})$$
Now we have $P(a,b)=P(3,5)$ , $A(a_1,b_1)=A(1,3)$ and $B(a_2,b_2)=B(7,9)$ :
$$\begin{array}{rl}& \Rightarrow P(a,b)=({\displaystyle \frac{k{a}_2+a_1}{k+1}},{\displaystyle \frac{k{b}_2+b_1}{k+1}})\Rightarrow a={\displaystyle \frac{k{a}_2+a_1}{k+1}}\\ & \Rightarrow 3={\displaystyle \frac{(k\times 7)+1}{k+1}}\Rightarrow 3={\displaystyle \frac{7k+1}{k+1}}\Rightarrow 3(k+1)=7k+1\\ & \Rightarrow 3k+3=7k+1\Rightarrow 3-1=7k-3k\Rightarrow 2=4k\\ & \Rightarrow k={\displaystyle \frac{1}{2}}\end{array}$$

Therefore, the ratio in which $$P(3,5)$$ divides $\overline{AB}$ is $1:2$ .

Note:
We can also solve this by following method with the graph, so we have:

Now, from the graph we have:
$\begin{array}{rl}& \Rightarrow x=(5-3)=2,y=(9-5)=4\\ & \Rightarrow x:y=2:4=1:2\\ & \Rightarrow a=(3-1)=2,b=(7-3)=4\\ & \Rightarrow a:b=2:4=1:2\end{array}$
Therefore, the ratio is $1:2$ .