Paragraph: Locus of the centroids of equilateral triangle inscribed in the parabola y^2=9 x is a conic section S=0 . On the basis of above information answer the following : Question: Length of latus rectum of S=0 is -

Step-by-step solution

y ^2=4 ax Let the centroid is ( h , k ) and vertices of are A ( at ₁^2, 2 at ₁ ), B ( at ₂^2, 2 at ₂ ), C ( at ₃^2, 2 at ₃ ) So 3 ~h = a ( t ₁^2+ t ₂^2+ t ₃^2 ) ...(i) 3 k =2 a ( t ₁+ t ₂+ t ₃ ) ...(ii) Let r is the radius of circle and t₁, t₂, t₃ are roots of equation (a t^2-h )^2+(2 a t-k)^2=r^2 a^2 t^4+t^2 (4 a^2-2 a h )-4 a k t+h^2+k^2-r^2=0t₁+t₂+t₃+t₄=0, t₁ t₂= 4 a^2-2 a h a^2 aligned & by (ii) t₄=- ( 3 k 2 a ) & by ( t ₁+ t ₂+ t ₃+ t ₄ )^2=0 & t ₁^2=-2 t ₁ t ₂ & 3 ~h a + ( 9 k ^2 4 a ^2 )=-2 ( 4 a ^2-2 ah a ^