Question

Draw the ellipse with the equation: 2𝑥 2 + 𝑦 2 + 4𝑥 − 9𝑦 = 7

Answer 1

first of all, we need to transform the equation to the canonical form

$\boxed{\frac {(x-x_0)^2} {a^2} + \frac{(y-y_0)^2} {b^2} = 1}$ - $x_0$ and $y_0$ - center coordinates

$2x^2 + 4x + y^2-9y=7$

$(2x^2 +4x +2) +(y^2 -9y+20.25) = 29.25$

$2(x+1)^2+(y-4.5)^2 = 29.25$

$\dfrac{(x+1)^2}{14.625} + \dfrac{(y - 4.5)^2}{29.25} = 1$

O(-1; 4.5) - center

$a = \sqrt{14.625} \thickapprox 3.8$ - minor semiaxis

$b = \sqrt{29.25} \thickapprox 5.4$ - major semiaxes