Identify each equation without completing the square.

Question

$9x^2+25y^2-54x-200y+256=0$

Accepted Answer

Hello, today we are going to be determining what type of conic our given equation represents without completing the square. So in order to approach this problem, we're going to need to use the generic conic. The generic conic is given to us as A X squared plus CY squared plus DX plus EY plus F is equal to zero. With this generic conic. There are four conditions that follow it. The first condition states that if the coefficients of the X squared and Y squared terms, which is A and C are equal to each other, then the conic represents a circle. If the term the coefficients a multiplied by C is equal to zero, then the conic represents a parabola. If the coefficients A and C are not equal to each other and their product is positive or greater than zero, then the conic represents an ellipse. And finally, if the product of the coefficients A and C are negative or less than zero, the conic represents a hyperbola. So what we need to do is we need to identify the coefficients in A and C in our given function or equation. If we take a look at the equation, we are given X squared plus nine Y squared minus 28 X minus 270 Y plus 2000, 185 is equal to zero. Currently, that is set up in the form of the generic conic. So we need to identify our A and C coefficient. Our A coefficient comes from the X squared term which in this case has a coefficient of one and our C coefficient comes from the Y squared term which is nine. So we know that A is equal to one and C is equal to nine already, we can see that A is not equal to C. So we can rule out the option of the conic being a circle. If we take the product A and C, we will get one multiplied by nine which is nine. This product is positive and greater than zero. Meaning that the conic given to us represents an ellipse. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding on how to approach this problem. And I'll go ahead and see you all in the next video.

Identify each equation without completing the square.
$9x^2+25y^2-54x-200y+256=0$

Key Concepts

Quadratic Equations

Conic Sections

Standard Form of Conic Sections

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