Question

If the normals at $(x_i,y_i)$, $i=1,2,3,4$ on the rectangular hyperbola, $xy=c^2$, meet at the point $(\alpha ,\beta )$, $\Sigma x_i=$

• A. $\frac{\alpha }{2}$
• B. $\alpha$
• C. $\beta$
• D. $\frac{\beta }{2}$

Answer 1

The correct option is B $\alpha$

Equation of normal to the rectangular hyperbola $xy=c^2$ at $t$ is given by,
$y-\frac{c}{t}=t^2(x-ct)$
$\Rightarrow ty=t^{3}x-c{t}^4+c$
Given it passes through $(\alpha ,\beta )$
$\Rightarrow t\beta =t^3\alpha -c{t}^4+c\Rightarrow c{t}^4-\alpha t^3+\beta t-c=0$
$\therefore {\sum }_i{t}_i=\frac{\alpha }{c}$
Thus ${\sum }_i{x}_i={\sum }_{i}c{t}_i=\alpha$
Hence, option 'B' is correct.