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 >3a cdot

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.

If a,b and c are distinct positive real numbers in A.P, then the roots of the equation ax^(2)+2bx+c=0 are

If a x^2+b x+c=0a n db x^2+c x+a=0 have a common root and a, b, and c are nonzero real numbers, then find the value of (a^3+b^3+c^3)//a b c

If y^2=4b(x-c) and y^2 =8ax having common normal then (a,b,c) is (a) (1/2,2,0) (b) (1,1,3) (c) (1,1,1) (d) (1,3,2)

If c is positive and 2a x^2+3b x+5c=0 does not have aby real roots, then prove that 2a-3b+5b>0.

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^3dot

The chord A B of the parabola y^2=4a x cuts the axis of the parabola at Cdot If A-=(a t_1 ^2,2a t_1),B-=(a t_2 ^2,2a t_2) , and A C : A B 1:3, then prove that t_2+2t_1=0 .