Question

If $a,b,c$ form a G.P. with common ratio $r;P$ and $Q$ are two points whose coordinates $(x,y)$ satisfy the relations $ax+by+c=0$ and $x^2-y^2-4=0$, then

• A. sum of the ordinates of $P$ and $Q$ is $\frac{2r^3}{1-r^2}$
• B. sum of the ordinates of $P$ and $Q$ is $\frac{2r^3}{r^2-1}$
• C. product of the ordinates of $P$ and $Q$ is $\frac{r^4-4}{r^2-1}$
• D. product of the ordinates of $P$ and $Q$ is $\frac{4-r^4}{r^2-1}$

Answer 1

Answer: (A), (C)

product of the ordinates of $P$ and $Q$ is $\frac{r^4-4}{r^2-1}$
sum of the ordinates of $P$ and $Q$ is $\frac{2r^3}{1-r^2}$

$ax+by+c=0\Rightarrow ax+ary+a{r}^2=0\Rightarrow x+ry+r^2=0\Rightarrow x=-(ry+r^2)$
Ordinates of $P$ and $Q$ are the roots of
$(r^2-1)y^2+2r^{3}y+r^4-4=0$
Hence sum of roots is $\frac{-2r^3}{(r^2-1)}$
and product of roots are $\frac{r^4-4}{(r^2-1)}.$