Identify the conic represented by the equation without completing...

Question

Identify the conic represented by the equation without completing the square. y^2 + 4x + 2y - 15 = 0

Accepted Answer

Hello today, we're going to identify what type of comic we have based off the non degenerate Konik given to us. So the problem states that we want to go ahead and identify this without using completing the square. So in order to do that, let's just go ahead and recall what the conditions of a non degenerate Connick is. While the form of a non degenerate Konik is technically a X squared plus C, Y squared plus D X plus E, Y plus F equal to zero. And there are four conditions that follow this. If A is equal to C then you have a circle. If the product of A&C are equal to zero you have a parabola. If A is not equal to C and eight times C is greater than zero, you have an ellipse and if A is not equal to C and eight times C is less than zero, you have a hyperbole. So in order to continue with this, we're gonna need to identify where our coefficients A and C are going to be. Well, the non degenerate Connor given to us is x squared plus four, Y squared plus two, X plus four, Y plus six equal to zero. And we want to specifically work with just our X squared and Y squared term because that's where A and C coefficient are gonna come from. So the a coefficient is just the term in front of the X squared term and that is just gonna be one. So A is going to equal to one and C is just gonna be the number in front of our y squared term and it's positive four. So C. Is going equal to four. Now, we can clearly see that A is not equal to C. So we can rule out that we have a circle. Well, all the other conditions require us to take the product of A. And C. So let's just go ahead and do that. So eight times C is going to be one times four and one times four is going to give us four. Now we know that four is greater than zero, and since the product of A times C is greater than zero and A is not equal to C. That means that we have an ellipse for iconic, and that means that the answer to this problem is going to be be So, I hope this video helps you in understanding how to find iconic without using completing the square. And I'll go ahead and see you all in the next video.

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