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

Answer: (A)

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