I have tried for more than hour now to solve this one.
If $4x^2+9y^2+z^2-6xy-3yz-2zx=0$ and an amount of $\rm Rs.3575/$- is distributed among three persons $\rm A,B,C$ in the ratio $\rm x:y:z$. Then the amount which is given to $\rm B$ is
$\rm A) \,\,Rs.600/$- $\qquad\rm B) \,\,Rs.750/$- $\qquad\rm C) \,\,Rs.650/$- $\qquad\rm D) \,\,Rs.700/$-
To solve for three variables (I only need B's amount though), we'll need three equations. However, I see only two in this question - the quadratic equation and then the equation from ratio and the total sum.$(x+y+z=3575)$
I thought that I could use the formula for $(a+b+c)^2$ as the third one, but that doesn't seem to do any good to me. I'm even starting to doubt if something is amiss with the question. I don't need the solution, I want to know the approach.
For the record, this is not homework. Shameful enough, I'm the volunteer teacher preparing a few kids for a competitive test and this question is from last year.
I would have expected to write the first equation as the sum of two squares. You could then take each square to be zero, giving you the two equations you seek. You can write $4x^2+9y^2+z^2-6xy-3yz-2zx=(x-z)^2+(x-3y)^2+2x^2-3yz$, but I can't see how to eat the $3yz$ term.