Problem 22 - Algebra 2

Question

Identify the conic section represented by each equation by writing the equation in standard form. For a parabola, give the vertex. For a circle, give the center and the radius. For an ellipse or a hyperbola, give the center and the foci. Sketch the graph. \(x^{2}+4 y^{2}-2 x-15=0\)

Step-by-Step Solution

Answer

The conic section is an ellipse with center at (1, 0), vertices at (3, 0) and (-1, 0), and foci at (1+/-\sqrt{3}, 0).

Step 1: Completing the square

Rewrite the given equation \(x^{2}+4 y^{2}-2 x-15=0\) in completed-square form: \[(x - 1)^2 + 4y^{2} = 16 .\] The general forms of standard equations are \( (x - h)^2 + (y - k)^2 = r^2\) for a circle and \[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\] for an ellipse.

Step 2: Identify the conic section and find its properties

Comparing the standardized equation \((x - 1)^2 + 4y^{2} = 16\) with the standard forms of the conic sections, it can be identified as an ellipse because it has different coefficients for \(x^2\) and \(y^2\), and both are positive. The standard form of an ellipse is \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\] Comparing, we find \(h = 1\), \(k = 0\), \(a^2 = 4\), \(b^2 = 16\), so \(a = 2\) and \(b = 4\). The foci of an ellipse are given by \((h ± ae, k)\), where \(e = \sqrt{1 - \frac{a^2}{b^2}} = \sqrt{1 - \frac{1}{4}} = \frac{\sqrt{3}}{2}\). So the foci are \((1±2\sqrt{3}/2, 0) = (1±\sqrt{3}, 0)\).

Step 3: Sketch the graph

Plot the center at (1, 0), the vertices along the x-axis at (1 - 2, 0) and (1 + 2, 0), and the foci at (1±\sqrt{3}, 0). Sketch the ellipse, which is elongated along the y-axis. Notice that the foci are inside the ellipse, and the distance from the center to each focus is smaller than the distance from the center to each vertex.

Key Concepts

Ellipse

An ellipse is a type of conic section that appears like a stretched circle. It is defined by two fixed points called foci. The sum of the distances from any point on the ellipse to the two foci is constant. The standard form of an ellipse's equation is \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]where

• \( (h, k) \) represents the center of the ellipse.
• \( a \) and \( b \) are the semi-major and semi-minor axes, respectively.
• If \( a > b \), the ellipse is longer along the x-axis, and if \( b > a \), it is longer along the y-axis.

The ellipse shows up in various practical applications such as planetary orbits and optics. Understanding its properties, such as the axes and foci, is crucial in sketching and analyzing its shape.

Completing the Square

Completing the square is a crucial algebraic technique used to manipulate quadratic equations into a form that reveals more information, often making it easier to solve or analyze them. This method is especially valuable in transforming an equation into one of the standard forms of conic sections. Here’s how it works:

• Take a quadratic term, for example \( x^2 - 2x \).
• To complete the square, you add and subtract the square of half the coefficient of \( x \), which is \( \left(\frac{-2}{2}\right)^2 = 1 \).
• This gives: \( x^2 - 2x + 1 - 1 \) which becomes \((x - 1)^2 - 1\).

Using this method, you can reformat the entire equation. This transformation helps you spot which type of conic section the equation represents, by aligning it to a recognizable standard form. It's like polishing a piece of metal so its true shine can be recognized.

Standard Form of Conic Sections

The standard form of conic sections helps categorize the type of curve represented by a quadratic equation. This form makes it easier to identify and sketch the graph of the equation. Each type of conic section has its own specific standard form:

• **Circle**: \( (x - h)^2 + (y - k)^2 = r^2 \)
• **Ellipse**: \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \)
• **Parabola**: \( y - k = a(x - h)^2 \) or \( x - h = a(y - k)^2 \)
• **Hyperbola**: \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \)

• In all formulas, \( (h, k) \) denotes the center or vertex.
• For an ellipse, \( a \) and \( b \) indicate the lengths of the axes.
• For hyperbolas and parabolas, these parameters help in determining the extent and direction of opening.

Knowing the standard form is key to easily transition from one format to another while analyzing equations. This framework lets you reveal the structure hidden behind complicated expressions, allowing for an accurate sketch of the graph.