Identifying Conic Sections

Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

x² + y² + 4x + 2y = 1

검증된 단계별 안내

Start with the given equation: \(x^2 + y^2 + 4x + 2y = 1\).

Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 + 2y) = 1\).

Complete the square for each group: - For \(x^2 + 4x\), take half of 4 (which is 2), square it (which is 4), and add inside the parentheses. - For \(y^2 + 2y\), take half of 2 (which is 1), square it (which is 1), and add inside the parentheses.

Since you added \(4\) and \(1\) inside the equation, add the same amounts to the right side to keep the equation balanced: \(1 + 4 + 1\).

Rewrite the equation in standard form using the completed squares: \[(x + 2)^2 + (y + 1)^2 = \text{new constant}\]. This form represents a circle, so identify the center and radius from this equation.

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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 geometric properties. By adding and subtracting appropriate constants, you transform terms like x² + 4x into (x + 2)² - 4. This technique is essential for rewriting conic section equations into standard forms.

Identification of Conic Sections

Conic sections include circles, ellipses, parabolas, and hyperbolas, each defined by specific standard equations. Recognizing the form of the equation after completing the square helps determine the type of conic, such as a circle if x and y terms have equal coefficients and the same sign.

Key Features of Conic Sections

Each conic section has characteristic elements: centers and vertices for ellipses and hyperbolas, foci for all conics, asymptotes for hyperbolas, and directrices for parabolas. Finding these features involves using the standard form of the conic and applying formulas related to distances and axes.

Identifying Conic SectionsComplete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.x² + y² + 4x + 2y = 1