If `p,q,r` are positive real number such that `(p^2+q^2)x^2-2q(p+r)x+q^2+r^2=0 and x^2-2x-8=0` has one root common then `(q^2+r^2)/p^2` is equal to

If p,q,r,s are natural numbers such that p>=q>=r If 2^(p)+2^(q)+2^(r)=2^(s) then

If the equations px^2+2qx+r=0 and px^2+2rx+q=0 have a common root then p+q+4r=

Prove that equations (q-r)x^(2)+(r-p)x+p-q=0 and (r-p)x^(2)+(p-q)x+q-r=0 have a common root.

If p,q,r>0 and p!=q!=r such that,f(x)=px^(2)+2qx+r,g(x)=qx^(2)+2rx+p,f(x)=0 and g(x)=0 has a common root.

If p, q, r each are positive rational number such tlaht p gt q gt r and the quadratic equation (p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0 has a root in (-1 , 0) then which of the following statement hold good? (A) (r + p)/(q) lt 2 (B) Both roots of given quadratic are rational (C) The equation px^(2) + 2qx + r = 0 has real and distinct roots (D) The equation px^(2) + 2qx + r = 0 has no real roots