Question

If the point $P(x_1+t(x_2-x_1),y_1+t(y_2-y_1\left)\right)$ divides the line segment joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$ internally, then the range of $t$ is

• A. $(-\infty ,0)$
• B. $(0,1)$
• C. $(1,\infty )$
• D. $\left\{1\right\}$

Answer 1

The correct option is B $(0,1)$

Let P divides $AB$ internally in the ratio $\lambda :1(\lambda >0)$
Now, using section formula we get
$\frac{x_1+\lambda x_2}{\lambda +1}=x_1+t(x_2-x_1)\Rightarrow \frac{x_1+\lambda x_2}{\lambda +1}-x_1=t(x_2-x_1)\Rightarrow \frac{\lambda (x_2-x_1)}{\lambda +1}=t(x_2-x_1)\Rightarrow t=\frac{\lambda }{\lambda +1}(\because x_1\ne x_2)\Rightarrow t=1-\frac{1}{\lambda +1}\therefore t\in (0,1)(\because \lambda >0)$