Question

All the chords of the hyperbola $3x^2-y^2-2x+4y=0,$ 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

The correct option is A $(1,-2)$

Let $ax+by=1$ be the chord ...(1)

Making the equation of hyperbola homogeneous using (1), we get $3x^2-y^2+(-2x+4y)(ax+by)=0$

$(3-2a)x^2+(-1+4b)y^2+(-2b+4a)xy=0$

Since the angle subtended at the origin is a right angle,so, coefficient of $x^2+$ coefficient of $y^2=0$

$\Rightarrow (3-2a)+(-1+4b)=0\Rightarrow a=2b+1$

$\therefore$ The chords are $(2b+1)x+by-1=0$

$\Rightarrow b(2+y)+(x-1)=0,$ which clearly pass through the fixed point $(1,-2)$.