The ratio in which the line joining points $A(-1,-1)$ and $B(2,1)$ divides the line joining $C(3,4)$ and $D(1,2)$

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Aicharya · Accepted Answer

$A(-1,-1), B(2,1)$ and $C(3,4), D(1,2)$ Then equation of line $A B$ $ \begin{aligned} y-y_1 & =\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right) \\ \Rightarrow \quad y+1 & =\frac{1+1}{2+1}(x+1) \\ \text { or } \quad 3 y+3 & =2 x+2 \text { or } 2 x-3 y-1=0 \end{aligned} $

Let the line $A B$ divides the line joining $C$ and $D$ in ratio $\lambda: 1$, then coordinates of $P$ are $ P=\left(\frac{\lambda+3}{\lambda+1}, \frac{2 \lambda+4}{\lambda+1}\right) $ Since, point $P$ lies on the line $A B$, putting it in equation of line $A B$, we obtain $ \begin{aligned} & & 2\left(\frac{\lambda+3}{\lambda+1}\right)-3\left(\frac{2 \lambda+4}{\lambda+1}\right)-1 & =0 \\ & \Rightarrow & 2 \lambda+6-6 \lambda-12-\lambda-1 & =0 \\ \Rightarrow & & -5 \lambda-7 & =0 \\ \Rightarrow & & \lambda & =-7 / 5 \\ & \therefore & \lambda: 1 & =7: 5 \end{aligned} $ Negative sign indicates that line $A B$ intersect it externally. $\therefore$ Desired Ratio $=7: 5$ externally

The ratio in which the line joining points $A(-1,-1)$ and $B(2,1)$ divides the line joining $C(3,4)$ and…

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