Ifa,b,c,p,q,r be six complex numbers such that `a/p+b/q+c/r=0`and `p/a+q/b+r/c=1-i` then prove that `(p^2/a^2)+(q^2/b^2)+(r^2/c^2)=-2i`

Let a, b , c, p , q , r be positive real numbers such that a , b , c are in G.P. and a^p = b^q = c^r . Then

Let , a , b ,c ,d , p ,q ,r be positive real number such that a ,b , c are in G.P and a^(p)=b^(q)=c^(r) Then-

If a/(q -r) = b/(r-p) = c/(p-q) , then prove that a + b + c = 0 =pa+ qb + rc .

If a, b, c, be respectively the pth, qth and rth terms of a G.P., prove that a^(q-r).b^(r-p).c^(p-q) = 1

If p=a+bomega+comega^2 , q=b+comega+aomega^2 , and r=c+aomega+bomega^2 , where a ,b ,c!=0 and omega is the complex cube root of unity, then (a) p+q+r=a+b+c (b) p^2+q^2+r^2=a^2+b^2+c^2 (c) p^2+q^2+r^2=-2(p q+q r+r p) (d) none of these

If p(q-r)x^2+q(r-p)x+r(p-q)=0 has equal roots, then prove that 2/q=1/p+1/rdot

If n_(p_(r-1))/a=n_(p_r)/b=n_(p_(r+1))/c ,then show that b^2=a(b+c)

Tangent is drawn at any point (p ,q) on the parabola y^2=4a xdot Tangents are drawn from any point on this tangant to the circle x^2+y^2=a^2 , such that the chords of contact pass through a fixed point (r , s) . Then p ,q ,r and s can hold the relation (a) r^2q=4p^2s (b) r q^2=4p s^2 (c) r q^2=-4p s^2 (d) r^2q=-4p^2s

If p+-a, q+- b, r+-c and abs((p,b,c),(a,q,c),(a,b,r)) = 0 show that p/(p-a)+q/(q-b)+r/(r-c)=2 .

If pth, qth, and rth terms of an A.P. are a ,b ,c , respectively, then show that (a-b)r+(b-c)p+(c-a)q=0