Problem 38 - Contemporary Precalculus

Question

Identify the conic section whose equation is given, and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$4 x^{2}+5 y^{2}-8 x+30 y+29=0$$

Step-by-Step Solution

Answer

Question: Identify the type of conic section represented by the equation \(4x^2 - 8x + 5y^2 + 30y = -29\), and find its properties such as center, foci, and vertices. Answer: The given equation represents an ellipse with center (1, -3), foci at (2, -3) and (0, -3), and vertices at \((1+\sqrt{5},-3)\) and \((1-\sqrt{5},-3)\).

Step 1: Complete the squares

To identify the ellipse's properties, we'll need to complete the squares for the equation. Re-arrange the terms to make it look like the general equation of an ellipse: $$4(x^2-2x)+5(y^2+6y)=-29$$ Now, complete the squares for x and y terms: $$4[(x-1)^2-1]+5[(y+3)^2-9]=-29$$ $$4(x-1)^2+5(y+3)^2=4+45-29$$ $$4(x-1)^2+5(y+3)^2=20$$

Step 2: Standard form

Now that we have completed the squares, we can write the equation in the standard form for an ellipse: $$\frac{(x-1)^2}{\frac{20}{4}}+\frac{(y+3)^2}{\frac{20}{5}}=1$$ $$\frac{(x-1)^2}{5}+\frac{(y+3)^2}{4}=1$$

Step 3: Properties of the ellipse

Now we can identify the properties of the ellipse: - Center: \((1, -3)\) - Semi-major axis (a): \(\sqrt{5}\) - Semi-minor axis (b): \(\sqrt{4}=2\) - Foci: Since it is a horizontal ellipse, the foci can be found using the formula \(c = \sqrt{a^2 - b^2}\), so \(c = \sqrt{5-4}=1\). The foci are located at \((1+c, -3)\) and \((1-c, -3)\), which gives us the points \((2, -3)\) and \((0, -3)\). - Vertices: Also, since it is a horizontal ellipse, the vertices can be found by adding and subtracting the value of the semi-major axis from the x-coordinate of the center which gives us \((1+\sqrt{5},-3)\) and \((1-\sqrt{5},-3)\). Now, we have all the required properties of the given ellipse.

Key Concepts

Ellipse

An ellipse is a special type of conic section that appears as an elongated circle. It's defined as the set of all points such that the sum of their distances from two fixed points, called foci, is constant. This unique property gives the ellipse its distinct oval shape.

An ellipse can be oriented either horizontally or vertically, which determines how it stretches along its axes.

• If the ellipse is wider than it is taller, it is horizontal.
• If it is taller than it is wide, it is vertical.

The line cutting through an ellipse from one side to the other is called the major axis, and it's the longest diameter of the ellipse. Perpendicular to this is the minor axis, the shortest diameter. Understanding these properties helps in graphing an ellipse and identifying its geometric features.

Completing the Square

Completing the square is a powerful algebraic technique used to transform quadratic expressions into a specific form that makes them easier to work with, especially when dealing with circles and ellipses.

This method involves adding and subtracting a term to each variable's quadratic expression to turn it into a perfect square trinomial. A perfect square trinomial is an expression that can be written as a binomial squared.

• For example, to complete the square for the expression \(x^2 - 2x\), you would add \(1\) (since \((\frac{-2}{2})^2 = 1\)) and revisit the expression as \((x-1)^2 - 1\).
• This allows you to reconfigure the original quadratic into a more manageable form.

Completing the square is essential when rewriting the equation of an ellipse to express it in its standard form, which reveals its inherent properties.

Standard Form of an Ellipse

The standard form of an ellipse equation is vital for identifying and understanding its features. This form appears as \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] where

• \((h, k)\) is the center of the ellipse,
• \(a\) represents the semi-major axis length,
• \(b\) represents the semi-minor axis length.

When the equation is in this structure, it's easy to determine the ellipse's center, axes lengths, and orientation (horizontal or vertical).

To achieve this equation, one often has to complete the square for the quadratic terms. For instance, if you start with a general quadratic equation, manipulating it into this format provides insight into the ellipse's geometric features, such as its center, vertices, and orientation.

Properties of an Ellipse

The properties of an ellipse are key to understanding how it is shaped and oriented. Once the equation is in standard form, these properties become quite apparent.

• Center: The point \((h, k)\) in the standard form, this is the midpoint between the foci.
• Semi-major axis: Represented by the larger of the values \(a\) or \(b\). This extends from the center to the ellipse's edge.
• Semi-minor axis: Represented by the smaller value, extending orthogonally from the center to the ellipse's edge.
• Foci: Located along the major axis, each foci is \(c\) units from the center, where \(c = \sqrt{a^2 - b^2}\).
• Vertices: These are the endpoints of the major axis, found \(a\) units away from the center along the major axis line.

Understanding these properties is crucial for graphing the ellipse and visualizing its dimensions and position in the plane.