Problem 24 - Algebra 2

Question

For Exercises \(23-25,\) match each equation below with the situation that it could represent. a. \(9 x^{2}+4 y^{2}-36=0\) b. \(0.004 x^{2}-x+y-3=0\) c. \(x^{2}+y^{2}-20 x+30 y-75=0\) PHOTOGRAPHY the oval opening in a picture frame

Step-by-Step Solution

Answer

Equation (a) matches the situation for an oval picture frame.

Step 1: Recognize Equation Forms

First, let's recognize the standard mathematical forms of the given equations. Equation (a) is of the form \(Ax^2 + By^2 + C = 0\), which resembles an ellipse. Equation (b) is a quadratic in \(x\) with all terms, possibly representing a parabola, and equation (c) is of the form \(x^2 + y^2 + Dx + Ey + F = 0\), which is usually a circle or can be rewritten to represent an ellipse.

Step 2: Match Equation to the Situation

Now we need to match the situation with the appropriate equation. An oval opening in a picture frame is typically represented by an ellipse. Therefore, we need an equation representing an ellipse. Equation (a) \(9x^2 + 4y^2 - 36 = 0\) can be rewritten as an ellipse of the form \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). Therefore, this equation corresponds to an ellipse.

Step 3: Verify the Match

Rewrite (a) into the standard form of an ellipse. Start by rearranging it: \(9x^2 + 4y^2 = 36\). Divide each term by 36, resulting in \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), confirming that it is indeed an ellipse, representing an oval shape, suitable for the opening in a picture frame.

Key Concepts

Standard Form of Ellipse

In geometry, an ellipse is a fascinating and essential shape. It extends our understanding of circles with its unique structure. The standard form of the equation of an ellipse is \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \]where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. This form helps categorize the ellipse's properties instantly.

In the exercise, equation (a) was rewritten from \(9x^2 + 4y^2 - 36 = 0\) to the form \[ \frac{x^2}{4} + \frac{y^2}{9} = 1, \] highlighting its elliptical nature. Note how we eliminate the constant term by rearranging and simplifying. Division by the constant term ensures that the equation equals 1, a critical step in revealing the standard form. The parameters \(a\) and \(b\) indicate the lengths along the x and y axes of the ellipse, providing a direct relation to its dimensions.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. The type of conic section results from the angle of the intersecting plane relative to the cone.

Ellipses occur when the angle of intersection creates a closed loop, which happens when a plane intersects the cone at an angle less than that of the cone's side. The equation that describes an ellipse usually has terms involving \(x^2\) and \(y^2\) with different coefficients, like in the example \(9x^2 + 4y^2 - 36 = 0\).

Conic sections are an essential part of geometry in algebra because they provide complex shapes described with simple equations. They bridge pure geometric shapes and algebraic representation, enabling profound analysis and solution of real-world problems.

Geometry in Algebra

Geometry and algebra converge to provide powerful tools for solving mathematical problems. This intersection, known as analytical geometry, uses algebraic equations to represent geometric shapes.

The benefit of this approach is that it brings precision and clarity, allowing for complex geometric shapes to be interpreted and analyzed algebraically. For example, recognising \(9x^2 + 4y^2 - 36 = 0\) as an algebraic representation of an ellipse allows us to deduce its shape and dimensions by transforming the equation into a standard form.

Through algebra, these geometric shapes can be manipulated, rotated, or resized within coordinate planes, thus broadening the theoretical and practical scope of both fields. Understanding how these equations translate into geometric figures enriches our knowledge and increases our capability to interpret and solve practical problems, such as identifying an object's shape, like an oval picture frame opening.