Shifting Conic Sections

Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.

9x² + 6y² + 36y = 0

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Rewrite the given equation to group the x and y terms: \(9x^{2} + 6y^{2} + 36y = 0\).

Complete the square for the y-terms to express the equation in standard form. First, factor out the coefficient of \(y^{2}\) from the y-terms: \(6(y^{2} + 6y)\). Then complete the square inside the parentheses by adding and subtracting the appropriate constant.

After completing the square, rewrite the equation in the form \(Ax^{2} + B(y - k)^{2} = C\), which will help identify the type of conic section (ellipse, hyperbola, or circle).

Divide through by the constant on the right side to normalize the equation to standard form, such as \(\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1\) for an ellipse or \(\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1\) for a hyperbola.

From the standard form, identify the center \((h, k)\), calculate the vertices and foci using the values of \(a\) and \(b\), find the equations of the asymptotes if it is a hyperbola, and determine the radius if it is a circle.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a form that reveals the conic's center or vertex. It involves adding and subtracting terms to create perfect square trinomials, which simplifies identifying key features like the center or radius.

Standard Forms of Conic Sections

Each conic section (circle, ellipse, parabola, hyperbola) has a standard equation form that highlights its geometric properties. Recognizing and converting the given equation into one of these forms helps determine the center, foci, vertices, asymptotes, and radius.

Properties of Ellipses and Circles

Understanding the definitions and properties of ellipses and circles, such as the relationship between the center, foci, vertices, and radius, is essential. These properties guide the identification of key points and distances once the conic is in standard form.

Shifting Conic SectionsFind the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.9x² + 6y² + 36y = 0