Cubic Equation in Two Variables Representing a Line.
The graph of the equation x 3 + y 3 − 9 x y + C = 0 contains a whole straight line for some real values of the constant C . Find the average of the maximum and the minimum value of C . If it is impossible to find that average enter 333.
The answer is 27.
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1 solution
I just realized that there is a simpler way of solving this problem. The graph of the equation x 3 + y 3 − 9 x y + c = 0 cannot contain a vertical line and it contains a line y = a x + b if and only if x 3 + ( a x + b ) 3 − 9 x ( a x + b ) + c = 0 for all real values of x . Using the undetermined coefficient method, we obtain that a = − 1 , b = − 3 , and c = 2 7 . So the answer must be 2 7 .
So, the level of this problem must be 3 or less.
We are going to assume that the graph of the equation x 3 + y 3 − 9 x y + C = 0 ( ∗ ) contains a straight line for certain value of C . If the equation contains a line, this line, of course, cannot be a vertical line. Then the given equation can define y as an implicit (linear) function of x and its derivative will be d x d y = − y 2 − 3 x x 2 − 9 y . Since the derivative of a linear function is constant, then the d x d y is constant at any point ( x , y ) satisfying the equation ( ∗ ) . Hence − y 2 − 3 x x 2 − 3 y = k ( ∗ ∗ ) for any point on that line. We can rearrange the last equation into the form ( x − 2 3 k ) 2 + k ( y − 2 k 3 ) 2 = 4 9 k 2 + 4 k 9 . The only case when this equation will be satisfied by all the points of a non-vertical line is when k is negative and 4 9 k 2 + 4 k 9 = 0 . Then we obtain that k = − 1 . Now substituting k = − 1 into the equation (**), we get − y 2 − 3 x x 2 − 3 y = − 1 , and this equation can be rewritten as ( x − y ) ( x + y + 3 ) = 0 , that is the equation of the intersecting lines y − x = 0 and x + y + 3 = 0 . Now it easy to see that the whole line y − x = 0 cannot be part of the graph of the equation ( ∗ ) , because there is no number C such that y − x divides x 3 + y 3 − 9 x y + C . Notice, that if you make y = x in the expression x 3 + y 3 − 9 x y + C , it becomes a polynomial that is not identically equal to 0. Therefore, we need to investigate if x + y + 3 = 0 is a factor of x 3 + y 3 − 9 x y + C for some value of C . Using division we obtain that this happens only when C = 2 7 . Additionally, x 3 + y 3 − 9 x y + 2 7 = ( x + y + 3 ) ( x 2 + y 2 − 3 x − 3 y − x y + 9 ) . Then the line x + y + 3 = 0 is contained in the graph of the equation ( ∗ ) . So the answer to the question is 2 7 .
By the way, as an interesting note, the only point on the graph of the equation ( ∗ ) that is not part of the line x + y + 3 = 0 is the pair of real numbers that is solution of x 2 + y 2 − 3 x − 3 y − x y + 9 = 0 . Using the quadratic formula you can see that this pair is ( 3 , 3 ) .