Question

I : The ratio in which $L=ax+by+c=0$ divides the line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ is $\frac{-L_1}{L_2}$( where $L_i=a{x}_i+b{y}_i+c$)
II: The equation of the line in which $(x_1,y_1)$ divides the line segment between the coordinate axes in the ratio $m:n$ is $\frac{nx}{x_1}+\frac{my}{y_1}=m+n$. Then which of the following is true?

• A. only I
• B. only II
• C. both I&II
• D. neither I nor II

Answer 1

The correct option is C both I&II

P lie on $L=ax+by+c=0$, So,

$a\frac{(k{x}_1+x_2)}{k+1}+b\frac{(k{y}_1+y_2)}{k+1}+c=0$

$k(a{x}_1+b{y}_1+c)+a{x}_2+b{y}_2+c=0$

$k=\frac{(-a{x}_2+b{y}_2+c)}{(a{x}_1+b{y}_1+c)}=\frac{-L_2}{L_1}$

Ratio in which $L=ax+by+c=0$ divides the line segment joining $A(x_1,y_1)B(x_2,y_2)$ is $\frac{-L_2}{L_1}$ True

$II.$ $(x_1,y_1)$ divides the line segment joining $(x,0)$ &$(0,y)$ in the ratio m:n

So, $x_1=\frac{nx}{m+n}$ & $y_1=\frac{my}{m+n}$

$x=\frac{x_1(m+n)}{n}$ & $y=\frac{y_1(m+n)}{m}$

So, equation of AB is $y=\frac{-y_{1}x}{n_{1}m}(x-\frac{x_1(m+n)}{n})$

$-my{x}_1=n{y}_{1}x-n_1{y}_{1}m-x_1{y}_{1}x$

$\Rightarrow \frac{hx}{x_1}+\frac{my}{y_1}=m+n$ True