Problem 51 - Precalculus: Mathematics for Calculus

Question

Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an cllipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$4 x^{2}+25 y^{2}-24 x+250 y+561=0$$

Step-by-Step Solution

Answer

The equation represents an ellipse centered at (3, -5).

Step 1: Identify the Conic Form

Start by organizing the given quadratic equation into standard conic form. The equation is \(4x^2 + 25y^2 - 24x + 250y + 561 = 0\). Notice that it contains both squared \(x\) and \(y\) terms with positive coefficients, suggesting it could be an ellipse. For clarity, rewrite it as \(4(x^2 - 6x) + 25(y^2 + 10y) = -561\).

Step 2: Complete the Square for x-Terms

Complete the square for the \(x\) terms. Start with \(x^2 - 6x\). Add and subtract \((6/2)^2 = 9\) inside the equation. Rewrite it as \((x-3)^2 - 9\). Substituting back, we have \(4((x-3)^2 - 9) = 4(x-3)^2 - 36\).

Step 3: Complete the Square for y-Terms

Now complete the square for the \(y\) terms. Start with \(y^2 + 10y\). Add and subtract \((10/2)^2 = 25\). Rewrite it as \((y+5)^2 - 25\). Substituting back, we get \(25((y+5)^2-25) = 25(y+5)^2 - 625\).

Step 4: Combine and Simplify the Equation

Substitute the results from completing the square back into the equation: \(4(x-3)^2 - 36 + 25(y+5)^2 - 625 = -561\). Simplify this equation: \(4(x-3)^2 + 25(y+5)^2 = 100\).

Step 5: Identify the Conic Type and Characteristics

Divide the entire equation by 100 to get \(\frac{(x-3)^2}{25} + \frac{(y+5)^2}{4} = 1\). This is the standard form of an ellipse centered at \((3, -5)\), with semimajor axis length 5 along the x-axis and semiminor axis length 2 along the y-axis. Since \(25 > 4\), the major axis is along the x-axis.

Step 6: Find Ellipse Properties

Identify the center, vertices, and foci of the ellipse. The center is at \((3, -5)\). The vertices are \((3 \, \pm 5, -5)\) or \((-2, -5)\) and \((8, -5)\). Find the foci \(c = \sqrt{25-4} = \sqrt{21}\); the foci are at \((3 \pm \sqrt{21}, -5)\) or approximately \((-1.58, -5)\) and \((7.58, -5)\).

Step 7: Sketch the Graph

Draw an ellipse centered at \((3, -5)\), elongated along the x-axis. Plot the vertices at \((-2, -5)\) and \((8, -5)\), and foci at approximations \((-1.58, -5)\) and \((7.58, -5)\). Indicate the semimajor axis of 5 and semiminor axis of 2.

Key Concepts

Ellipse

An ellipse is a type of conic section that appears as a flattened or elongated circle. It has several key characteristics that distinguish it from other conic sections, such as parabolas and hyperbolas. An ellipse results from the intersection of a plane with a cone at an angle that is not perpendicular to the cone's base and is less steep than the side of the cone.

• The major axis is the longest diameter of the ellipse, passing through its center and the two furthest points on the perimeter.
• The minor axis is the shortest diameter, which intersects the major axis at the ellipse's center.
• The foci (plural for focus) are two fixed points located along the major axis such that the total distance from any point on the ellipse to the foci remains constant.

Understanding these features is crucial for solving problems involving ellipses. Recognizing that an equation represents an ellipse involves identifying both squared variables ( x i.e., x² and y²) with positive coefficients, and determining the relative axis lengths.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a form that makes them easier to interpret, often transforming them into a standard geometric form. This technique is particularly useful for converting quadratic equations of conic sections, like ellipses, into their respective standard forms. Consider the expression x² + bx. To complete the square, follow these steps:

• Find half of the coefficient of x, then square it. For x² - 6x, take (-6/2)² = 9.
• Add and subtract this squared number within the expression to preserve equality: x² - 6x becomes (x - 3)² - 9.

By completing the square for both the x and y terms in an equation, we can reorganize the equation into a more recognizable notation. This facilitates determining its characteristics, such as the center and axes for ellipses.

Equation of an Ellipse

The standard equation of an ellipse takes one of these forms depending on the orientation of its axes: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]When the major axis is horizontal, or:\[\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\]When the major axis is vertical.In both cases:

• (h, k) represents the center of the ellipse.
• a is the semi-major axis length, indicating whether the ellipse is wider or taller.
• b is the semi-minor axis length.

In our exercise, the equation \(\frac{(x-3)^2}{25} + \frac{(y+5)^2}{4} = 1\)shows an ellipse centered at (3, -5) with a horizontal major axis, because its larger denominator (25) relates to \(x\).Understanding this equation helps us identify key attributes like the vertices at (3±5,-5) and the foci, which can be calculated using \(c = \sqrt{a^2 - b^2}\).

Graphing Ellipses

Graphing an ellipse requires plotting its basic elements based on the equation parameters, aiding in visual understanding. Begin by identifying the center of the ellipse, which provides a reference for positioning other features like the axes and foci.

• From the center, move in both directions along the major and minor axes to find the vertices. The points are key for outlining the shape of the ellipse. In our example, the center is (3,-5), and the vertices along the x-axis are (-2, -5) and (8, -5).
• Locate the foci using the derived equation \(c = \sqrt{a^2 - b^2}\). This provides further insight into the ellipse’s configuration and symmetry. For the exercise, the foci are (-1.58,-5) and (7.58,-5).

With these elements plotted, sketch smooth curves connecting the vertices to form the ellipse, ensuring symmetry about both the major and minor axes. This graphical representation solidifies understanding of the ellipse's structure and its corresponding geometric properties.