Multiple Choice
How can you slice a vertically oriented 3D cone with a 2D plane to get a circle?
8. Conic Sections
Introduction to Conic Sections
3:00 minutesProblem 54
Textbook Question
Identify each equation without completing the square.
Verified step by step guidance1
Recognize the given equation: \(9x^2 + 4y^2 - 36x + 8y + 31 = 0\). This is a quadratic equation in two variables, which suggests it might represent a conic section.
Identify the coefficients of \(x^2\) and \(y^2\). Here, the coefficients are 9 and 4, respectively. Since both coefficients are positive and different, the equation represents an ellipse.
To confirm it's an ellipse, check if the equation can be rewritten in the standard form of an ellipse: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). This involves completing the square for both \(x\) and \(y\) terms.
Notice that the equation has linear terms \(-36x\) and \(8y\). These terms will be used in the process of completing the square, which is not required here but indicates the presence of a center \((h, k)\).
Conclude that the equation represents an ellipse based on the analysis of the coefficients and the structure of the equation, without needing to complete the square.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations
A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0, where a, b, and c are constants. In the given equation, the presence of x^2 and y^2 terms indicates that it is a quadratic in two variables. Understanding how to manipulate and analyze these equations is crucial for identifying their properties and solutions.
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Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The equation provided can represent different conic sections, such as ellipses or hyperbolas, depending on the coefficients and the discriminant. Recognizing the type of conic section helps in understanding the geometric properties and the graph of the equation.
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Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, which makes it easier to solve or analyze. Although the question specifies not to complete the square, understanding this technique is essential for rewriting quadratic equations in vertex form, which can reveal important characteristics such as the vertex and direction of opening.
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