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

Question

Identify the conic represented by the equation without completing the square. 4x^2 - 9y^2 - 8x + 12y - 144 = 0

Accepted Answer

Hello. Today we're going to be distinguishing what type of Connick we have without using completing the square. So the Konik given to us is four, X squared minus Y squared over six plus eight X plus three Y is equal to zero. Now before we continue with this let's just go ahead and recall what the coefficients mean. So this equation is in the form of a non degenerate Connick and the form of Ananda djukanovic is typically a X squared plus C, Y squared plus D X plus E, Y Plus F equal to zero. And the conditions of this is that if your coefficients A is equal to see you have a circle, If a times C is equal to zero you have a parabola. If A is not equal to C and eight times C is less than zero, then you have a hyperbole to and if A is not equal to C and a times C is greater than zero, you have an ellipse. So in order to figure out what type of comic we have without completing the square, we need to go ahead and first identify what are A and C coefficients are in this equation. Okay, well, the A and C coefficients are just the coefficients in front of the x squared term and the y squared term. So let's go ahead and just write out what our x squared and y squared term is. So in this equation are x square term is going to be four X squared and r y squared term is negative Y squared over six. Now I am gonna go ahead and rewrite this term very quickly because negative y squared over six is the same thing as same negative 1/6 times y squared. So now we just want to go ahead and list out what the coefficients of these values are. While the coefficient of our x squared term is going to be a is equal to four and the coefficient of our white square term is going to be C is equal to negative 1/6. Now, what we want to do is go ahead and see if these are equal to each other, while four is not the same thing as negative 1/6. So A is not equal to C. And that means that we don't have a circle. Well, all the other conditions require that we take the product of a N C. So let's figure out what A times C is. Well, eight times C is the fourth same thing as four times negative 1/6 and four times negative 1/6 is gonna give us negative 4/6, which simplifies to negative 2/3. Well, now we have the value of the product of a N C. And we know that a negative number is always going to be less than zero. So since a is not equal to C. And the product of a N C is less than zero. That means that we're going to have a hyperbole for this Connick and that's going to be the solution to this problem, and the answer to the problem is going to be C. So I hope this video helps you in understanding how to identify iconic without completing the square. And I'll go ahead and see you all in the next video.

Identify the conic represented by the equation without completing the square. 4x^2 - 9y^2 - 8x + 12y - 144 = 0

Key Concepts

Conic Sections

Standard Form of Conic Equations

Discriminant of Conic Sections

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