The condition that parabola y 2 = 4 a x & y 2 = 4 c ( x − b ) have a common normal other than x-axis is ( a  = b  = c )

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The normal to y 2 = 4 a x at t is y = − t x + 2 a t + a t 3 , and the normal of the same slope to y 2 = 4 c ( x − b ) is y = − t x + b t + 2 c t + c t 3 . They coincide when 2 a t + a t 3 = b t + 2 c t + c t 3 , i.e. t 2 ( a − c ) = b − 2 ( a − c ) , so t 2 = a − c b ​ − 2 . A common normal other than the x-axis exists exactly when t 2 > 0 , i.e. a − c b ​ > 2 .

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