# How do you know if #x^2-10x-y+18=0# is a hyperbola, parabola, circle or ellipse?

This is a parabola.

Given:

We can also complete the square to find:

graph{(x^2-10x-y+18)(x-5+0.0001y)((x-5)^2+(y+7)^2-0.01) = 0 [-6.455, 13.545, -7.88, 2.12]}

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To determine the type of conic section represented by the equation (x^2 - 10x - y + 18 = 0), we can examine the coefficients of (x^2), (x), and (y).

A hyperbola is characterized by equations with terms involving both (x^2) and (y^2) with opposite signs. A parabola typically has either a (x^2) term or a (y^2) term, but not both. A circle has both (x^2) and (y^2) terms with the same coefficient. An ellipse has both (x^2) and (y^2) terms with different coefficients, but the signs are the same.

In the given equation, there is only one squared term, which is (x^2). This indicates that the conic section may be either a parabola or a circle.

To confirm, we can check the coefficients of the linear terms. The coefficient of (x) is -10, indicating that the axis of symmetry is vertical if it's a parabola. However, if it's a circle, it doesn't provide enough information.

Therefore, without additional information, we cannot definitively determine whether the conic section represented by the given equation is a parabola or a circle.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- Find the equation of directrix and focus of the parabola (x-2)^2=8(y+1)?

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