Hyperbola Asymptotes and Conic Classification

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Calculate hyperbola asymptotes and identify conic sections from general second-degree equations.

Hyperbola Asymptotes and Conic Classification

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Find the equations of the asymptotes for: $$\frac{y^2}{25} - \frac{x^2}{4} = 1$$

$$y = \pm \frac{5}{2}x$$

What is the general second-degree equation for any conic section in the Cartesian plane?

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

If $B = 0$ in $Ax^2 + Cy^2 + Dx + Ey + F = 0$, how do the signs of $A$ and $C$ distinguish an ellipse from a hyperbola?

Ellipse: $A$ and $C$ have the same sign ($AC > 0$) with $A \neq C$. Hyperbola: $A$ and $C$ have opposite signs ($AC < 0$).

If $B = 0$ in $Ax^2 + Cy^2 + Dx + Ey + F = 0$, what condition identifies a circle rather than a non-circular ellipse?

$A = C \neq 0$, and both have the same sign.

What formula defines the conic section discriminant?

$$\Delta = B^2 - 4AC$$

How does the discriminant $B^2 - 4AC$ classify non-degenerate conic sections?

Ellipse or circle: $B^2 - 4AC < 0$ Parabola: $B^2 - 4AC = 0$ Hyperbola: $B^2 - 4AC > 0$

How does the numerical eccentricity $e$ categorize circles, ellipses, parabolas, and hyperbolas?

Circle: $e = 0$ Ellipse: $0 < e < 1$ Parabola: $e = 1$ Hyperbola: $e > 1$

How do the relationships for finding focal distance $c$ differ fundamentally between an ellipse and a hyperbola?

In an ellipse, vertices are farther from center than foci: $$c^2 = a^2 - b^2$$ In a hyperbola, foci are farther from center than vertices: $$c^2 = a^2 + b^2$$

What degenerate shapes can form instead of a standard ellipse or hyperbola?

An ellipse can degenerate into a single point or no graph (empty set). A hyperbola can degenerate into two intersecting straight lines.

What are the equations of the asymptotes for a horizontal hyperbola centered at $(h, k)$?

$$y - k = \pm \frac{b}{a}(x - h)$$

What are the equations of the asymptotes for a vertical hyperbola centered at $(h, k)$?

$$y - k = \pm \frac{a}{b}(x - h)$$

What algebraic shortcut gives the equations of the asymptotes directly from the hyperbola's standard form?

Replace the constant $1$ with $0$ on the right side of the standard equation, factor the difference of squares, and solve for $y - k$.

Find the equations of the asymptotes for: $$\frac{x^2}{9} - \frac{y^2}{16} = 1$$

$$y = \pm \frac{4}{3}x$$

Hyperbola Asymptotes and Conic Classification

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