Problem 41 - Algebra 2

Question

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 5 x^{2}+6 x-4 y=x^{2}-y^{2}-2 x $$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Simplify the Equation

Rearrange the equation by moving all terms to one side: \( 5x^2 + 6x - 4y - x^2 + y^2 + 2x = 0 \), simplifying to \( 4x^2 + 8x + y^2 - 4y = 0 \).

Step 2: Identify the Degree and Structure

Notice that the highest degree terms are \(4x^2\) and \(y^2\), indicating the equation includes squared terms but not both variables squared with their combination. There's no term like \(xy\), and the coefficients in front of \(x^2\) and \(y^2\) are positive.

Step 3: Classify Based on Terms and Signs

Equations of conic sections can be determined by the presence and signs of squared terms. Since both \(x^2\) and \(y^2\) terms have positive coefficients and no mixed \(xy\) term, it signifies these do not represent a parabola or hyperbola. Ellipses require both squared terms with the same sign, but usually, different coefficients, indicating the graph is an ellipse.

Key Concepts

Parabola

A parabola is a unique conic section that can be visualized as a "U"-shaped curve. It arises when you cut through a cone parallel to one of its sides. The equation of a parabola typically has one squared term, either for the variable \(x\) or \(y\), but not both together in the same equation.

For instance:

• If the squared term is \(x^2\), you might see an equation like \(y = ax^2 + bx + c\).
• If the squared term is \(y^2\), it might be written as \(x = ay^2 + by + c\).

The absence of both squared terms together (such as \(x^2\) and \(y^2\)) indicates the shape of a parabola, distinguishing it from other conic sections. This form is notably present in pathways such as projectile motions.

Circle

A circle is a simple and symmetrical shape. It's the set of all points that are equidistant from a centered point. For identification, its equation includes both \(x^2\) and \(y^2\) with equal coefficients.

Here is what a standard circle equation looks like:

• \((x - h)^2 + (y - k)^2 = r^2\),

where \((h, k)\) represents the center and \(r\) denotes the radius. If both squared terms have the same coefficient and there is no \(xy\) term, the graph will be circular.

It's fascinating to note how this uniform equation represents a perfect symmetry horizontally and vertically.

Ellipse

An ellipse looks like a stretched circle. It's often described as an oval or egg-like shape. The equation of an ellipse will involve both \(x^2\) and \(y^2\), typically with different coefficients, reflecting its oval nature.

Consider this basic form:

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

Here, \((h, k)\) is the center of the ellipse, and \(a\) and \(b\) are distances from the center to the ellipse along the \(x\) and \(y\) axes, respectively. If \(a = b\), then the ellipse is a circle. For our exercise, having no \(xy\) term and positive coefficients for both squared terms indicates the graph is an ellipse.

Hyperbola

A hyperbola is another intriguing conic section, characterized by its distinct "double-curved" look. It features two mirror-opposite curves. The general form of a hyperbola equation includes differences between squares of \(x\) and \(y\).

Typically, a standard equation looks like this:

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

Deviations from this can show as ellipse-like equations, but with opposite signs between the squared terms.

A hyperbola reflects a unique open-ended shape, unlike the closed shape of a circle or an ellipse. It remains distinct from other conic sections through its characteristic structure and properties, such as its asymptotes.