Question

The equation of the normals to the rectangular hyperbola $xy=4$ which are parallel to the line $2x-y=5$ are given by

• A. $2x-y-3\sqrt{2}=0$
• B. $2x-y-2\sqrt{2}=0$
• C. $2x-y+3\sqrt{2}=0$
• D. $2x-y+2\sqrt{2}=0$

Answer 1

Answer: (A), (C)

The correct options are
A $2x-y-3\sqrt{2}=0$
C $2x-y+3\sqrt{2}=0$

Parametric equation of rectangular hyperbola $xy=4$ are given by $x=2t,y=\frac{2}{t}$

$\therefore$ Slope of normal at a point ${}^{\prime }{t}^{\prime }=-\frac{dx}{dy}$ $=-\left(\frac{dx/dt}{dy/dt}\right)=-\left(\frac{2}{-2{/t}^2}\right)=t^2$

$A.T.Q:t^2=2\Rightarrow t=\pm \surd 2$

and the point are $(2\surd 2,\surd 2)$ and $(-2\surd 2,-\surd 2)$

Equation of normals are

$(y-\sqrt{2})=2(x-2\surd 2)$ and $(y+\sqrt{2})=2(x+2\surd 2)$

$\Rightarrow 2x-y-3\surd 2=0$ and $2x-y+3\surd 2=0$