Find real solution to the equation3x2+3y2−4xy+10x−10y+10=0.What does partial differentiation give for a second degree equation which doesn't represent a conic?

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Find real solution to the equation3x2+3y2−4xy+10x−10y+10=0.What does partial differentiation give for a second degree equation which doesn&#039;t represent a conic?

ocretz56 · Accepted Answer

Step 1  A conic curve, is the intersection of the 3D cone  p1z2=p2x2+p3y2  with the 3D plane  p4x+p5y+p6=0  which takes the general form as stated in the OP. Now, a point can also be regarded as a conic curve (or conic, simply) in a sense that it is the intersection of the 3D plane with the vertex of the 3D cone, i.e. the point in which, the two halves of the cone meet.  An example is the intersection of  z2=x2+y2 with 2z=x+y  Hence, your approach is correct. Another way for justifying the result, is to note that any single point is an ellipse with zero diameters and ellipse is a conic curve.

Gordon Stephens · Accepted Answer

Step 1Multiplying by 3 then &quot;completing the square&quot; for the quadratic in x:9x2+9y2−12xy+30x−30y+30=9x2−6(2y−5)x+(2y−5)2−(2y−5)2+9y2−30y+30=(3x−(2y−5))2+5y2−10y+5=(3x−2y+5)2+5(y−1)2It follows that the only real solution is y=1, x=−1Step 2The final equation can be written asx′2+y′2=0 with x′=3x−2y+5 and y′=5y−5,which is a degenerate conic, specifically two intersecting complex lines x′±iy′=0 with the single common real point at x′=y′=0. This is consistent with OP&#039;s findings of discriminant ≤0 and center of the conic at the unique real point.

Find real solution to the equation 3x^{2}+3y^{2}-4xy+10x-10y+10=0. What does partial differentiation give

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