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Manipal MET2016MathematicsHyperbola

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

Options

  1. A(1,-2)
  2. B(-1,2)
  3. C(1,2)
  4. DNone of these

Correct answer

A. (1,-2)

Step-by-step solution

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. Coefficient of x^2+ Coefficient of y^2=0 array ll & (3-2 a)+(-1+4 b) & =0 & a=2 b+1 array 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) .

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