Yes, the x-squared is missing. Also, the xy term is missing. In my book the general form of a quadratic equation in 2 variables is
.
After a Google Search, I have found this equation also expressed as
,
which is closer to what you have.
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Yes.
Actually, now that I think about it, if there is an xy term then the conic is rotated by some angle. You stated that you're starting to learn about conic sections, so the book probably omitted the xy term deliberately. This means that in parabolas & hyperbolas, without the xy term, the curves will only open upward/downward/left/right. And in ellipses, the major/minor axes will only be horizontal/vertical lines.
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