Question

If x-axis divides the line joining $(3,-4)$ and $(5,6)$ in the ratio $a:b$, then what is the value of $\frac{a}{b}$?

• A. $\frac{3}{2}$
• B. $\frac{2}{3}$
• C. $\frac{3}{4}$
• D. $\frac{1}{3}$

Answer 1

The correct option is B $\frac{2}{3}$

Given that, the $x-axis$ divides the line joining $(3,-4)$ and $(5,6)$ in the ratio $a:b$.

To find out: The value of $\frac{a}{b}$

Let $P$ be any point on the $x-axis$ having coordinates $(x,0)$

We know that, if a point $P(x,y)$ divides a line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$ internally, then the coordinates of the point $P$ are:

$x=\frac{m{x}_2+n{x}_1}{m+n}$ and $y=\frac{m{y}_2+n{y}_1}{m+n}$

Here, $m=a,n=b,x_1=3,y_1=-4,x_2=5,y_2=6$

$\therefore (x,0)=(\frac{5a+3b}{a+b},\frac{6a-4b}{a+b})$

Comparing the $y-coordinate$, we get:

$\frac{6a-4b}{a+b}=0$

$\Rightarrow 6a-4b=0$

$\Rightarrow 6a=4b$

$\Rightarrow \frac{a}{b}=\frac{4}{6}$

$\therefore \frac{a}{b}=\frac{2}{3}$

Hence, if the $x-axis$ divides the line joining $(3,-4)$ and $(5,6)$ in the ratio $a:b$, then $\frac{a}{b}=\frac{2}{3}$.