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We have the quadratic equation:

$$ax^2 + bx + cy^2 + dy + e$$ $a$ and $c$ are both negative or both positive.

How can I, by looking at that only, determine whether $b$ (the length of the semi-minor axis) will be bigger than $a$ (the length of the semi major axis)?

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What is the equation? You've given an expression... –  Sanath Apr 19 at 18:29
You're using $a$ and $b$ with two different meanings... –  Hans Lundmark Apr 19 at 19:37
And by the way, the length of the semi-minor axis of an ellipse is by definition never greater than the length of the semi-major axis, so you don't even need to look at anything. –  Hans Lundmark Apr 19 at 19:39

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