Problem 53 - Precalculus

Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$9 x^{2}+4 y^{2}-18 x+16 y-119=0$$

Step-by-Step Solution

Answer

The graph of the given equation is an ellipse.

Step 1: Group x and y Terms

First, arrange the equation by grouping x terms and y terms together: \(9x^{2}-18x+4y^{2}+16y-119=0\).

Step 2: Complete the Square for x and y

To complete the square, take the coefficients of the x and y terms, divide by two and square it. For x terms, \((-18/2(9))^2=1\) and for y terms, \((16/2(4))^2=4\). Adding these into the equation gives us \(9(x^{2}-2x+1)+4(y^{2}+4y+4)=119+9+16\).

Step 3: Simplify the Equation

This simplifies to: \(9(x-1)^{2}+4(y+2)^{2}=144\). Divide entire equation by 144 to get it in standard form. This gives us \(\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{36}=1\).

Step 4: Identify the Conic Section

The standard form \(\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{36}=1\) is an equation of ellipse, because it is in the form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\).

Key Concepts

Ellipse

An ellipse is one of the conic sections that represents an elongated circle, characterized by its distinct oval shape. It is defined mathematically as the set of all points such that the sum of the distances from two fixed points, called foci, is constant. Imagine stretching a circle; that's essentially how an ellipse forms.

Ellipses have two main axes: the major axis, which is the longest diameter running through the center and both foci, and the minor axis, which is the shortest diameter perpendicular to the major axis. Understanding an ellipse's axes helps in visualizing its shape and orientation.

Some practical examples of ellipses include the orbits of planets, which due to gravitational forces, often exhibit slight elliptical paths rather than perfect circles.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a perfect square trinomial. This method is particularly useful when dealing with conic sections, like ellipses, to help simplify and solve their equations.

The process involves taking the coefficient of the linear term (like \(x\) or \(y\) terms), dividing it by two, and then squaring the result. This squared value is then used to re-write the quadratic part of the equation as a perfect square.

For example, if you have \(9x^2 - 18x\), you'd focus on the \(-18x\). Half of \(-18/9\) is 1, and squaring it gives 1. Add this inside the equation to get a perfect square, so \((x-1)^2\).

Completing the square thus facilitates rewriting complex equations into simpler forms, making them easier to solve and analyze.

Equation of an Ellipse

An ellipse's equation in standard form is essential for identifying its properties and understanding its orientation. The standard form of an ellipse with a horizontal major axis is expressed as \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\). Conversely, if the major axis is vertical, the form is \((x-h)^2/b^2 + (y-k)^2/a^2 = 1\).

In these equations, \(h, k\) represent the center of the ellipse, while \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. When \(a > b\), the major axis is horizontal, and when \(b > a\), it is vertical.

Understanding the equation of an ellipse allows us to determine key features like its center, foci, and axes. These insights are crucial for applications in physics and astronomy, where ellipses frequently model real-world phenomena.