Let $\alpha,\beta$ be the roots of the equation $ax^2+2bx+c=0$ and $\gamma,\delta$ be the roots of the equation $px^2+2qx+r=0$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then-

If a, b, c, d are in GP, then b/a = c/b = d/c = r. Roots are a, ar, ar^2, ar^3. From ax^2+2bx+c=0, sum = a+ar = -2b/a and product = a^2r = c/a. From px^2+2qx+r=0, sum = ar^2+ar^3 = -2q/p and product = a^2r^5 = r/p. Dividing products: (a^2r^5)/(a^2r) = (r/p)/(c/a) => r^4 = (r/p)*(a/c). This leads to the relation q^2ac = b^2pr.