JEE Main2006MathematicsParabolaActual

The locus of the vertices of the family of parabolas y= a^3 x^2 3 + a^2 x 2 -2 a is

Options

Ax y= 105 64
Bx y= 3 4
Cx y= 35 16
Dx y= 64 105

Correct answer

A. x y= 105 64

Step-by-step solution

Parabola: y= a^3 x^2 3 + a^2 x 2 -2 a Vertex: ( , ) aligned & = -a^2 / 2 2 a^3 / 3 =- 3 4 a , = - ( a^4 4 +4 a^3 3 2 a ) 4 a^3 3 =- - ( 1 4 + 8 3 ) a^4 4 3 a^3 & =- 35 12 a 4 3=- 35 16 a & =- 3 4 a (- 35 16 ) a= 105 64 . aligned

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