If the parabolas `y^2=4a x` and `y^2=4c(x-b)` have a common normal other than the x-axis `(a , b , c` being distinct positive real numbers), then prove that `b/(a-c)> 2.`

If a , b ,c are three distinct positive real numbers in G.P., then prove that c^2+2a b >3ac

The condition that the parabolas y^2=4c(x-d) and y^2=4ax have a common normal other than X-axis (agt0,cgt0) is

If a,b,c are three distinct positive real numbers in G.P., than prove that c^2+2ab gt 3ac .

If the parabols y^(2) = 4kx (k gt 0) and y^(2) = 4 (x-1) do not have a common normal other than the axis of parabola, then k in

Parabolas (y-alpha)^(2)=4a(x-beta)and(y-alpha)^(2)=4a'(x-beta') will have a common normal (other than the normal passing through vertex of parabola) if

Column I, Column II If a ,b ,c ,a n dd are four zero real numbers such that (d+a-b)^2+(d+b-c)^2=0 and he root of the equation a(b-c)x^2+b(c-a)x=c(a-b)=0 are real and equal, then, p. a+b+c=0 If the roots the equation (a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0 are real and equal, then, q. a ,b ,ca r einAdotPdot If the equation a x^2+b x+c=0a n dx^3-3x^2+3x-0 have a common real root, then, r. a ,b ,ca r einGdotPdot Let a ,b ,c be positive real number such that the expression b x^2+(sqrt((a+b)^2+b^2))x+(a+c) is non-negative AAx in R , then, s. a ,b ,ca r einHdotPdot

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x >3.

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x 3

Find the angle between the parabolas y^2=4a x and x^2=4b y at their point of intersection other than the origin.

If the length of a focal chord of the parabola y^2=4a x at a distance b from the vertex is c , then prove that b^2c=4a^3 .