Problem 65 - Precalculus with Limits

Question

In Exercises 57-72, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(4x^2+3y^2+8x-24y+51=0\)

Step-by-Step Solution

Answer

The given equation represents an ellipse once it's rearranged into the standard form.

Step 1: Recognize the given equation

The equation provided is a second degree polynomial equation in x and y, which indicates that it represents a conic section. But to identify which conic section it represents, we need to put it in the standard form.

Step 2: Rearrange the given equation

Rearrange the given equation by grouping the x terms and the y terms together: \(4(x^2+2x)+3(y^2-8y)=-51\).

Step 3: Complete the square for x and y

Completing the square allows us to get our equation into a standard form. To complete the square, we need to add and subtract the square of half of coefficient of x and y in the bracket. The rearranged equation then turns into: \(4[(x+1)^2-1]+3[(y-4)^2-16]=-51\). By simplifying this, we get: \(4(x+1)^2+3(y-4)^2=3\).

Step 4: Identify the standard form

When simplified, it is clear that this equation fits into the standard form of an ellipse, \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\), where (h,k) is the center of the ellipse and a and b are the semi-major and semi-minor axes.

Key Concepts

Ellipse

An ellipse is a beautiful, oval-shaped curve that you might have come across in everyday life, such as the shape of a running track or the path of planets around the sun. In mathematics, an ellipse can be described by its standard form equation:

• \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)

Here,

• \( (h, k) \) represents the center of the ellipse.
• \( a \) and \( b \) determine the length of the ellipse's axes.

The longest diameter of the ellipse is known as the major axis, while the shortest is the minor axis. These axes are perpendicular to each other and intersect at the center.

The special aspect of an ellipse is that the sum of the distances from any point on the ellipse to two fixed points, called foci, remains constant. This property gives ellipses their characteristic shape and involves fascinating mathematical properties.

Second Degree Polynomial Equations

Second degree polynomial equations in two variables, like \( x \) and \( y \), form a broad class of equations that represent various shapes known as conic sections. These include ellipses, parabolas, hyperbolas, and circles.

Typically, these equations take the general form:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

The presence and arrangement of the terms \( Ax^2 \) and \( Cy^2 \), as well as the coefficient values, help to classify the conic sections:

• Ellipses have both \( x^2 \) and \( y^2 \) with the same sign (and not equal coefficients).
• Circles are a special case of ellipses where \( A = C \) and \( B = 0 \).
• Parabolas have either \( x^2 \) or \( y^2 \) but not both.
• Hyperbolas have \( x^2 \) and \( y^2 \) with opposite signs.

By converting these general forms into their respective standard forms, it is easier to visualize the graph of the equation.

Completing the Square

Completing the square is a valuable algebraic technique used to simplify quadratic equations and transform them into a form that is easier to analyze, particularly helpful for converting quadratic equations into the standard forms of conic sections.

• For a quadratic equation \( ax^2 + bx + c \), completing the square involves finding a perfect square trinomial.
• To do this, take half of the linear coefficient \( b \), square it, and add and subtract that value within the expression.

This technique is instrumental in rewriting second degree polynomial equations in their readily recognizable standard forms.
For example, if you have \( x^2 + 6x \), then:

• Half of 6 is 3, and \( 3^2 = 9 \).
• Add and subtract 9: \( (x^2 + 6x + 9 - 9) \).
• This becomes \( (x+3)^2 - 9 \).

Once in this form, it becomes much easier to identify the properties of ellipses, parabolas, and other related curves.