Geometrical interpretation of P(x)+Q(y)=0 when P, Q are polynomials of degree 2?

Fahdvfm

Fahdvfm

Answered question

2022-11-02

Geometrical interpretation of P ( x ) + Q ( y ) = 0 when P,Q are polynomials of degree 2?

Answer & Explanation

cismadmec

cismadmec

Beginner2022-11-03Added 22 answers

By completing the square, you find that
( a x 2 + b x + c ) + ( d y 2 + e y + f ) = ( a ( x α ) 2 + s ) + ( d ( y β ) 2 + t ) )
so that your equation has the form:
a ( x α ) 2 + d ( y β ) 2 = κ .
Assume first that a , d > 0 (or both are negative). If κ < 0, then there are no solutions, while if κ = 0, then the only solution is the point ( α , β ). The interesting case is when κ > 0. Then you get a circle if a = d and an ellipse otherwise, centered at the point ( α , β )

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