Problem 74 - Precalculus with Limits

Question

CAPSTONE Consider the equation \(6x^2-3xy+6y^2-25=0\). (a) Without calculating, explain how to rewrite the equation so that it does not have an \(xy\)-term. (b) Explain how to identify the graph of the equation.

Step-by-Step Solution

Answer

In order to get rid of the xy term, the equation can be written as \(6(\frac{x}{2})^2-3\frac{x}{2}y+6y^2-25=0\). Then we substitute \(p = \frac{x}{2}\). To identify the graph, we look at the coefficients: positive \(x^2\) and \(y^2\) terms after expanding, indicating a circle graph since the coefficients are equal.

Step 1: Rewrite the Equation Without the xy-Term

To remove the xy-term in the equation, view the equation as a quadratic in x. \[6(\frac{x}{2})^2-3\frac{x}{2}y+6y^2-25=0.\]Then perform a substitution where \(p = \frac{x}{2}\). Then replace x with \(2p\) in the equation and simplify, effectively removing the xy term.

Step 2: Return Substituted Variable to Original

After removing the xy term and simplifying, bring back the \(x\) variable by substituting \(x = 2p\) back into the equation.

Step 3: Determine the shape of the Graph

The shape of the graph can be determined by the signs of the coefficients in the equation. A positive \(x^2\) term and a positive \(y^2\) term suggests that the graph will be an ellipse. If the coefficient of the \(x\) term is equal to the coefficient of the \(y\) term, it indicates that the ellipse is a circle.

Key Concepts

Ellipse

An ellipse is an important concept in conic sections. It's similar to a stretched circle. If you think about how a circle is perfectly round, an ellipse is basically an oval, flattened version of it.
You can often identify an ellipse from its equation because it will have both an \(x^2\) and a \(y^2\) term. In our modified equation from the exercise, the coefficients of \(x^2\) and \(y^2\) were both positive, typical of an ellipse. No \(xy\) term in the final equation also confirmed that the graph is an ellipse.
Understanding ellipses can lighten your load when graphing. You check if these terms have the same coefficient; if they do without a cross-product term, it's a circle. But if they're different and still positive, off you go into ellipse-land. Ellipses have two axes of symmetry and are characterized by two focal points. The sum of distances from these focal points to any point on the ellipse remains constant.

Quadratic Equation

Quadratic equations typically show up in the form \(ax^2 + bxy + cy^2 + dx + ey + f = 0\). The term \(bxy\) is often what makes these equations trickier, especially when you're trying to determine the conic they're describing.
In the exercise, we performed a clever substitution to eliminate the \(xy\)-term, which simplified our equation into a standard format. This format hints that the equation represents a conic section, and knowing how to manipulate it is key.
The coefficients \(a\), \(b\), and \(c\) play a crucial role in identifying the conic section shape from a quadratic equation. By observing these coefficients, you can spot whether it's an ellipse, hyperbola, or parabola. For ellipses, you just need \(a\) and \(c\) to be positive and \(b\) should be zero when the equation is in its simplest form.

Coordinate Geometry

Coordinate geometry, or analytic geometry, blends algebra with geometry to describe geometric figures using coordinates. This approach allows you to study conics like ellipses by analyzing equations instead of only drawing shapes.
In the day-to-day context of geometry, equations help us visualize where points lie and what shape they form. With the revised equation from the exercise, using substitution allows easier interpretation of the shape through coordinates, making understanding ellipses straightforward.
When working with coordinate geometry, always remember to:

• Identify variables and their roles in the equation.
• Confirm the absence of cross-products or align them if necessary.
• Determine the positivity or negativity of squared term coefficients.

With these steps, determining graphs and conic sections becomes as straightforward as drawing them. Mastery of analytic geometry can provide a clearer look into the deeper geometrical nature of algebraic equations.