13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


25y² - 4x² = 100

Verified step by step guidance

1. Rewrite the given equation \(25y^{2} - 4x^{2} = 100\) in standard form by dividing both sides by 100 to isolate the terms: \(\frac{25y^{2}}{100} - \frac{4x^{2}}{100} = 1\).
2. Simplify the fractions to get \(\frac{y^{2}}{4} - \frac{x^{2}}{25} = 1\). This is the standard form of a hyperbola centered at the origin with the \(y^{2}\) term positive, indicating it opens vertically.
3. Identify the values of \(a^{2}\) and \(b^{2}\) from the equation: \(a^{2} = 4\) and \(b^{2} = 25\). Here, \(a\) corresponds to the distance from the center to each vertex along the \(y\)-axis, and \(b\) relates to the shape of the hyperbola.
4. Find the vertices of the hyperbola, which lie along the \(y\)-axis at \((0, \pm a)\), so the vertices are at \((0, \pm 2)\).
5. Calculate the focal distance \(c\) using the relationship \(c^{2} = a^{2} + b^{2}\), then find the coordinates of the foci at \((0, \pm c)\). Finally, write the equations of the asymptotes, which for a vertical hyperbola centered at the origin are \(y = \pm \frac{a}{b} x\).

Key Concepts

Identification of Conic Sections from General Equations

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The general second-degree equation Ax² + By² + Cx + Dy + E = 0 can represent a parabola, ellipse, or hyperbola depending on the signs and values of A and B. If A and B have opposite signs, the conic is a hyperbola; if they are equal and nonzero, it is a circle; if both are positive but unequal, it is an ellipse; if one variable is squared and the other is linear, it is a parabola.

Standard Forms and Key Features of Hyperbolas

A hyperbola's standard form is (x-h)²/a² - (y-k)²/b² = 1 or (y-k)²/a² - (x-h)²/b² = 1, where (h,k) is the center. Key features include vertices (points where the hyperbola intersects its principal axis), foci (points inside the branches related to the definition of the hyperbola), and asymptotes (lines the hyperbola approaches but never touches). These features help in sketching and understanding the hyperbola's shape.

Graphing and Analyzing Conic Sections

Graphing conic sections involves rewriting the equation in standard form, identifying the center, vertices, foci, and axes lengths, and plotting these points. For hyperbolas, asymptotes are found using slopes ±b/a or ±a/b depending on orientation. Labeling these elements accurately provides a clear visual representation and aids in understanding the conic's geometric properties.