Paragraph: Tangents are drawn from a variable point P to the ellipse x^2+4 y^2=4 meeting it at Q & R . So Q R is chord of contact of pair of tangents drawn from P . On the basis of above information answer the following : Question: If the sum of inclination of PQ and PR is constant ( 90^ ) , then the locus of P is a -

Step-by-step solution

Ellipse x^2 4 +y^2=1 Let P ( h , k ) and equation of any tangent is aligned & y=m x 4 m^2+1 (k-m h)^2=4 m^2+1 & ~m ^2 ( ~h ^2-4 )-2 mkh + ( k ^2-1 )=0 & ₁+ ₂= & ( ₁+ ₂ )= & m ₁+ m ₂ 1- m ₁ ~m ₂ = 2 hk ~h ^2-4 1- k ^2-1 ~h ^2-4 = aligned aligned & 2 xy = ( x ^2- y ^2-3 ) & x ^2-2 xy - y ^2=3 & =3-0-0-0+3( )^2 0 aligned H ^2 AB so hyperbola