Question

The ratio which divides the line joining the points $(2,3)$ and $(4,2),$ also divides the segment joining the point $(1,2)$ and $(4,3)$ is

• A. $1:2$
• B. $(3:1)$
• C. $(1:4)$
• D. $1:1$

Answer 1

The correct option is D $1:1$

Let $CD$ be divided in the ratio $a:b$ by AB

Equation of line with two points

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

$\Rightarrow AB\Rightarrow y-3=\frac{1-3}{4-2}(x-2)$

$\Rightarrow x+y=5$ ………(i)

and $CD\Rightarrow y-2=\frac{3-2}{4-1}(x-1)$

$\Rightarrow x-3y=-5$ ………(ii)

From (i) & (ii)

We get $y=\frac{5}{2},x=\frac{+5}{2}$

We have to find the ratio $a:b$

We get P as $(\frac{5}{2},\frac{5}{2})$

then $(\frac{5}{2},\frac{5}{2})=(\frac{b+4a}{a+b},\frac{2b+3a}{a+b})$

$\Rightarrow \frac{4a+b}{a+b}=\frac{5}{2}$ and $\frac{5}{2}=\frac{2b+3a}{a+b}$

$\Rightarrow 8a+2b=5a+5b$

$\Rightarrow 3a=3b$

$\Rightarrow \frac{a}{b}=1$.