Find domain and codomain for relation given by ellipse equation.

Question

Given a relation between $x$ and $y$ defined by equation: $$x^2 -2x +y^2 +xy = 20$$

How does one determines its domain and codomain?

Answer 1

Hint: Write $$4x^2-8x+4y^2+4xy =80$$ so $$(2y+x)^2+3x^2-8x =80\;\;\;\;/\cdot 3\;\;\;/+16$$ so

$$3(2y+x)^2 +9x^2-24x+16 = 256$$

or $$3(2y+x)^2 +(3x-4)^2 = 256$$

Does this help? You can find a range for $x$ from here and then a range for $y$.

Answer 2

Consider your equation first as a quadratic equation for $x$, then as an equation for $y$: the discriminant condition $\Delta\ge0$ will give you the required ranges.

Equation for $x$: $$ x^2+(y-2)x+y^2-20=0 $$ which gives: $$ \Delta=-3y^2-4y+84. $$ We have $\Delta\ge0$ for $-6\le y\le{14\over3}$, which is the range of $y$.

Arrange then as an equation for $y$ to find the range of $x$.