Question

All the chords of the hyperbola $3 x^{2}-y^{2}-2 x+4 y=0$, subtending a right angle at the origin pass through the fixed point

• A. $(1,-2)$
• B. $(-1,2)$
• C. $(1,2)$
• D. None of these

Answer 1

Answer: (A)

Let $a x+b y=1$ be the chord ...(i)
Making the equation of hyperbola homogeneous using Eq. (i), we get
$3 x^{2}-y^{2}+(-2 x+4 y)(a x+b y)=0$
or $(3-2 a) x^{2}+(-1+4 b) y^{2}+(-2 b+4 a) x y=0$
Since, the angle subtended at the origin is a right angle
$\therefore $ Coefficient of $x^{2}+$ Coefficient of $y^{2}=0$
$\Rightarrow(3-2 a)+(-1+4 b)=0 $
$\Rightarrow a=2 b+1$
$\therefore $ Chords are $(2 b+1) x+b y-1=0$
or $b(2+y)+(x-1)=0$
which clearly pass through the fixed point $(1,-2)$