For a National Board Exam Review:
Compute the eccentricity of a given curve $9x^2 + 4y^2 - 24y + 144 = 0$
Answer is $0.75$
I try:
$$9x^2 + 4y^2 - 24y + 144 = 0$$ $$9x^2 + 4(y^2 - 6y + 9) = -144 + 36$$ $$9x^2 + 4(y-3)^2 = -108$$ $$\frac{-x^2}{12} - \frac{4(y-3)^2}{27} = 1$$
For an ellipse:
$$ a^2 - b^2 = c^2 $$ $$ 12 - 27 = 15 $$
$$ e = \frac{c}{a} = \frac{ \sqrt{15} }{ \sqrt{12} } = \frac{\sqrt{5}}{2} $$
It's also odd because I have never encountered a conic which has both terms negative. What am I doing wrong?