In Exercises 49–56, identify each equation without completing the...

Question

In Exercises 49–56, identify each equation without completing the square.y^2 - 4x + 2y + 21 = 0

Accepted Answer

Hello, everyone in this video, we're gonna be looking at this practice problem where we have the equation X squared minus two X minus four, Y plus 29 is equal to zero. And we want to determine if that equation is a circle. A parabola, an ellipse or a hyperbola. And in order to determine that we need to remember some of the properties of each of these. So when we have a non degenerate conic section with the form A X squared plus CY squared plus DX plus EY plus F equals zero where A and C are not both zero, we have a circle. If A is equal to C, we have a problem if A, a times C equals zero on ellipse, if A does not equal C and A times C is greater than zero. And then finally, a hyperbola if A times C is less than zero. So we're gonna use these properties to help us determine what our equation is. And the first step is to determine the A value and the C value. So in this case, the A value is the coefficient in front of X squared. And for ours, we have a coefficient of one. So A is equal to one and C is the coefficient in front of the Y squared. And our equation does not have a Y squared. So C is equal to zero. So from here you can see that the condition for the circle is for A to equal C and for us A does not equal C. So the circle is eliminated and the condition for the parabola is for a time C to equal zero. And for us, a time C equals zero. So it could be a problem. Let's check the other ones just in case. So the condition for the ellipse is for a two, not equal zero. And for a time C to be greater than zero. And the first condition is true where A does not equal C however, fails a second condition since A times C is zero. So it's not an ellipse. And then finally, a hyperbola, the condition is for a time C to be less than zero and it fails once again, since a time C is zero. Exactly. So you can see that the problem is the only one that fits the conditions. So this becomes our final answer and that aligns with answer choice B. So hopefully this video helped and see you next time.

In Exercises 49–56, identify each equation without completing the square.y^2 - 4x + 2y + 21 = 0

Key Concepts

Quadratic Equations

Standard Form of a Conic Section

Completing the Square

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