Question 973011
Let the roots of x^2+bx+c=0  be g and h, and
Let the roots of x^2+qx+r=0 be m and n

Let ratio of roots be given by g/h = R1 and m/n = R2

Now, g+h = -b ---(i)
and gh =c ---(ii)
Similarly,
m+n = -q ---(iii)
mn= r ----(iv)
Squaring eqn (i),
{{{g^2+h^2+2gh = b^2}}}
=> {{{g^2+h^2 = b^2-2c}}} (using eqn. (ii)-----(v)
Divinding(v) by (ii)
{{{(g^2+h^2)/gh = (b^2-2c)/c}}}
=> {{{(g/h)+(h/g) = (b^2-2c)/c}}}
=> {{{R1+1/R1 =(b^2-2c)/c}}}
Similarly,
{{{R2+1/R2 =(q^2-2r)/r}}}
Since R1=R2 (as per ques.),
So, {{{R1+1/R1 = R2+1/R2}}}
=> {{{(b^2-2c)/c=(q^2-2r)/r}}}
=> {{{b^2r-2cr = q^2c-2cr}}}
=> {{{b^2r = q^2c}}}
Hence option (iv) is correct.