If `G` is the geometric mean of `xa n dy` then prove that `1/(G^2-x^2)+1/(G^2-y^2)=1/(G^2)`

If a ,b ,a n dc are in G.P. then prove that 1/(a^2-b^2)+1/(b^2)=1/(b^2-c^2)dot

If a curve is represented parametrically by the equation x=f(t) and y=g(t)" then prove that "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(3)((d^(2)x)/(dy^(2)))

If a, b and c are in G.P and x and y, respectively , be arithmetic means between a,b and b,c then prove that a/x+c/y=2 and 1/x+1/y=2/b

If f(x)=x+tanxa n df is the inverse of g, then g^(prime)(x) equals (a) 1/(1+[g(x)-x]^2) (b) 1/(2-[g(x)-x]^2) (c) 1/(2+[g(x)-x]^2) (d) none of these

If the normal at P(theta) on the hyperbola (x^2)/(a^2)-(y^2)/(2a^2)=1 meets the transvers axis at G , then prove that A GdotA^(prime)G=a^2(e^4sec^2theta-1) , where Aa n dA ' are the vertices of the hyperbola.

If m is the A.M. of two distinct real numbers l and n""(""l ,""n"">""1) and G1, G2 and G3 are three geometric means between l and n, then G1 4+2G2 4+G3 4 equals, (1) 4l^2 mn (2) 4l^m^2 mn (3) 4l m n^2 (4) 4l^2m^2n^2

If a ,a_1, a_2, a_3, a_(2n),b are in A.P. and a ,g_1,g_2,g_3, ,g_(2n),b . are in G.P. and h s the H.M. of aa n db , then prove that (a_1+a_(2n))/(g_1g_(2n))+(a_2+a_(2n-1))/(g_1g_(2n-1))++(a_n+a_(n+1))/(g_ng_(n+1))=(2n)/h

if m is the AM of two distinct real number l and n and G_1,G_2 and G_3 are three geometric means between l and n , then (G_1^4 + G_2^4 +G_3^4 ) equals to

f is a strictly monotonic differentiable function with f^(prime)(x)=1/(sqrt(1+x^3))dot If g is the inverse of f, then g^(x)= a. (2x^2)/(2sqrt(1+x^3)) b. (2g^2(x))/(2sqrt(1+g^2(x))) c. 3/2g^2(x) d. (x^2)/(sqrt(1+x^3))