Question

If the normal to the rectangular hyperbola xy = c${}^2$ at the point $(ct,\frac{c}{t})$ meets the curve again at $(c{t}^{\prime },\frac{c}{t^{\prime }}),$then

• A. t³t^' =1
• B. t³t^' = - 1
• C. tt^' =1
• D. tt^' = - 1

Answer 1

Answer: (B)

The equation of tangent at $(ct,\frac{c}{t})$ is
ty= $t^{3}x-c{t}^4+c$
if it passes through $(c{t}^{\prime },\frac{c}{t^{\prime }})$then
$\Rightarrow \frac{tc}{t^{\prime }}=t^{3}c{t}^{\prime }-c{t}^4+c$
$\Rightarrow t=t^3{t}^{{}^{\prime }2}-t^4{t}^{\prime }+t^{\prime }$
$\Rightarrow t.t^{\prime }=t^3{t}^{\prime }(t^{\prime }.t)\Rightarrow t^3{t}^{\prime }=-1$
Note : If we take the co-ordinate axes along the asymptotes of a rectangular hyperbola, then the general equation $x^2-y^2=a^2$ becomes xy = $c^2,$ where c is a constant.