Question

Points $\left(\frac{1}{2},\frac{-13}{4}\right)$ divides the line joining the points $\left(3,-5\right)$ and $\left(-7,2\right)$ in what ratio

• A. $1:3$ internally
• B. $3:1$internally
• C. $1:3$ externally
• D. $3:1$externally

Answer 1

The correct option is A

$1:3$ internally

Explanation for the correct option

Finding the ratio in which the line joining the two points is divided.

According to the Section formula, we know that if the point $x=(a,b)$ intersects the line joining the two points $x_1=\left(a_1,b_1\right)$ and $x_2=\left(a_2,b_2\right)$ in the ratio $m:n$ internally, then the point $x=(a,b)$ is given by the following :

$a=\frac{m{a}_2+n{a}_1}{m+n}$ and $b=\frac{m{b}_2+n{b}_1}{m+n}$.

Now, $\left(\frac{1}{2},\frac{-13}{4}\right)$ must lies on the straight line joining the two points $\left(3,-5\right)$ and $\left(-7,2\right)$. So, $\left(\frac{1}{2},\frac{-13}{4}\right)$ intersects the line internally in some ratio.

Assuming the ratio to be $k:1$.

So, $m=k\&n=1$.

Also, $x_1=\left(3,-5\right)$ and $x_2=\left(-7,2\right)$

Putting the values $m=k\&n=1$ in the above formula, we get,

$$\begin{gathered} \frac{1}{2}=\frac{\left(-7k+3\right)}{(k+1)} \\ \Rightarrow k+1=-14k+6 \\ \Rightarrow k+14k=6-1 \\ \Rightarrow 15k=5 \\ \Rightarrow k=\frac{1}{3} \end{gathered}$$

Hence, the required ratio is $\frac{1}{3}:1$ i.e. $1:3$.

Hence, the correct answer is Option (A).