I notice that other problems are answered in a timely manner....was is my overlooked?

Determine values for A, B, and C such that the equation below represents the given type of conic. Each axis of the ellipse, parabola, and hyperbola should be horizontal or vertical. Then rewrite your equation for each conic in standard form, identify (h, k), and describe the translation.

Ax^2+By+Cy^2+2x-4y-5=0

Part A: Circle
Part B: Ellipse
Part C: Parabola
Part D: Hyperbol

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Ms. Sue,

The same exact question was just asked today from another student and WAS answered in a timely manner. My however is still blank...... :(
Good thing is I have my answer now.....

To determine values for A, B, and C such that the equation represents each given type of conic, let's analyze the general form of the equation:

Ax^2 + By + Cy^2 + 2x - 4y - 5 = 0

We'll go through each part of the question to find the values for A, B, and C, rewrite the equation in standard form, identify (h, k), and describe the translation.

Part A: Circle

To represent a circle, we want A, B, and C to have the same value since both x^2 and y^2 terms should be present. Additionally, there should be no linear terms (x or y) and a constant term (without x or y). The equation in standard form for a circle is:

(x - h)^2 + (y - k)^2 = r^2

To rewrite the equation for a circle, we need to complete the square for both x and y terms:

x^2 + 2x + y^2 - 4y = 5

To complete the square, we need to add and subtract the square of half the coefficient of x and y, respectively:

x^2 + 2x + 1 + y^2 - 4y + 4 = 5 + 1 + 4

Now, we can rewrite the equation:

(x + 1)^2 + (y - 2)^2 = 10

Therefore, for the equation to represent a circle, A = B = C, and the equation in standard form is (x + 1)^2 + (y - 2)^2 = 10. The center of the circle is (-1, 2), and there is no translation.

Part B: Ellipse

To represent an ellipse, we also want A and C to have the same value (coefficients of x^2 and y^2 terms). However, there can be linear terms (2x and -4y). Again, there should be a constant term present. The equation in standard form for an ellipse is:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

To rewrite the equation for an ellipse, we need to complete the square for both x and y terms:

x^2 + 2x + y^2 - 4y = 5

To complete the square, we need to add and subtract the square of half the coefficient of x and y, respectively:

x^2 + 2x + 1 + y^2 - 4y + 4 = 5 + 1 + 4

Now, we can rewrite the equation:

(x + 1)^2 + (y - 2)^2 / 4 = 10

Therefore, for the equation to represent an ellipse, A = C, and the equation in standard form is (x + 1)^2 + (y - 2)^2 / 4 = 10. The center of the ellipse is (-1, 2), and there is no translation.

Part C: Parabola

To represent a parabola, the equation should only contain terms of the same power (either x or y, not both). Additionally, there should be no xy term. The equation in standard form for a parabola is:

(x - h)^2 = 4p(y - k)

To rewrite the equation for a parabola, we need to identify if x or y terms dominate. In this case, x terms dominate since there are both x^2 and x terms. Let's complete the square for x:

x^2 + 2x + y^2 - 4y = 5

To complete the square for x, we need to add and subtract the square of half the coefficient of x:

x^2 + 2x + 1 + y^2 - 4y = 5 + 1

Now, we can rewrite the equation:

(x + 1)^2 + y^2 - 4y = 6

Therefore, for the equation to represent a parabola, there should be no A or C terms present, and the equation in standard form is (x + 1)^2 + y^2 - 4y = 6. The vertex of the parabola is (-1, 2), and there is no translation.

Part D: Hyperbola

To represent a hyperbola, we want A and C to have opposite signs. Additionally, there should be linear terms present (2x and -4y). The equation in standard form for a hyperbola is:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

To rewrite the equation for a hyperbola, we need to separate the x and y terms and complete the square for each:

x^2 + 2x + y^2 - 4y = 5

To complete the square for both x and y, we need to add and subtract the square of half the coefficient:

x^2 + 2x + 1 + y^2 - 4y + 4 = 5 + 1 + 4

Now, we can rewrite the equation:

(x + 1)^2 - (y - 2)^2 / 4 = 10

Therefore, for the equation to represent a hyperbola, A and C should have opposite signs, and the equation in standard form is (x + 1)^2 - (y - 2)^2 / 4 = 10. The center of the hyperbola is (-1, 2), and there is no translation.