Question

The value of $\prod x_i=\prod y_i$ is equal to :

• A. $-c^2$
• B. $-a^2$
• C. $-b^2$
• D. $-c^4$

Answer 1

The correct option is D $-c^4$

Let $P(x_1,y_1)=(ct,\frac{c}{t}),(i=1,2,3,4)$ are points on the rectangular hyperbola $xy=c^2$
and equation ofnonnal to the hyperbola $xy=c^2$ at $(x_1,y_1)$
is $x{x}_1-y{y}_1=x_1^2-y_1^2$ so replacing
$x_1\to ct$ and $y_1\to \frac{c}{t}$
we get equation of normal given by
$xct-\frac{yc}{t}=c^2{t}^2-\frac{c^2}{t^2}$
$\Rightarrow x{t}_3-yt-c{t}^4+c=0$ which passes through the points $(a,b)$
$\therefore c{t}^4-a{t}^3+bt-c=0$ which is fourth degree equation in $t$, so in general 4 normals can be drawn as the fourth degree equation has four roots. Let $t_1,t_2,t_3,t_4$ be the roots of fourth degree equation, then
$\sum t_i=\frac{a}{c}\sum t_1{t}_2=0,\sum t_1{t}_2{t}_3=\frac{-b}{c}and{t}_1{t}_2{t}_3{t}_4=-1$
$\prod x_i=c^4\prod t_i=c^4(-1)=-c^4$
and $\prod y_i=c^4\left(\frac{1}{\prod t_i}\right)=c^4\left(\frac{1}{-1}\right)=-c^4$
$\therefore \prod x_i=\prod y_i$
Hence, option 'D' is correct.