If 3x2+2αxy+2y2+2ax-4y+1 can be resolved into two linear factors, prove that α is the root of the equation x2+ 4ax+2a2+6=0. please don't solve the problem just hint is expected.
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Originally Posted by sumedh If 3x2+2αxy+2y2+2ax-4y+1 can be resolved into two linear factors, prove that α is the root of the equation x2+ 4ax+2a2+6=0. Hint : The discriminant of the second degree equation on : must be a perfect square.
That is, if you meant "can be resolved into two linear factors" with rational coefficients.
Originally Posted by HallsofIvy That is, if you meant "can be resolved into two linear factors" with rational coefficients. Why?. The problem is equivalent to find where the given conic is degenerated. That is i.e. . Why do we need rational coefficients ?
on solving i got α^2 y^2+a^2+2aαy =-6y^2-3+12y [α means alpha] α^2 y^2+a^2+2aαy -9 (y-3)^2 after equating what should i do??? could you please tell me the concept for solving this????
Last edited by sumedh; Aug 17th 2011 at 08:40 AM. Reason: edit
The discriminant of the equation is then, is a perfect square iff the discriminant of is . Now, verify . Equivalently, is a root of .
thank you very much i got it