Question

If the points $A(a,b)$, $B(-a,-b)$ and $P(a^2,ab)$ are collinear then the ratio in which $P$ divides $\overline{AB}$ is

• A. $1+a:1-a$
• B. $1:a$
• C. $a:1$
• D. $1-a:1+a$

Answer 1

The correct option is A $1+a:1-a$

Let P divides AB in $1:m$

$A(a,b),B(-a,-b)$

$\therefore a^2=\frac{-a-am}{1-m}\Rightarrow a=\frac{1+m}{m-1}$

$ab=\frac{-b-bm}{1-m}\Rightarrow a=\frac{1+m}{m-1}$

$\therefore a=\frac{1+m}{m-1}=1+\frac{2}{m-1}$

$\therefore a-1=\frac{2}{m-1}$

$m-1=\frac{2}{a-1}$

$\therefore m=\frac{2+a-1}{a-1}$

$=\frac{a+1}{a-1}$

$\therefore$ Ratio $a+1:a-1$