<span>1. circle
2. parabola
3. ellipse
4. hyperbola
Let's look at each equation and see what they are:
1. x^2 + y^2 - 4x + 6y - 5 = 0
* There's a lot of crud in this equation, but the thing to note is that the x^2 and y^2 terms have the same scaling factor (which is 1). This should scream "circle" to you.
2. x^2 - 6y = 0
* Key thing to note here is that the y term isn't squared, but the x term is squared. This is a key sign that the equation is a parabola.
3. 4x^2 + 9y^2 = 1
* Here we have the sum of an x squared term and a y squared term. That kinda sounds like a circle, but there's those 2 coefficients scaling the results. And they're different. So we're looking for a closed curve that kinda looks like a circle, but it's stretched out a bit. And that's an ellipse.
4. 7x^2 - 9y^2 = 343
* Here we have a couple of squared terms for x and y. But we're not adding them together, we're subtracting. And that indicates a hyperbola.</span>
