Textbook Question
In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes.
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8. Conic Sections
Hyperbolas NOT at the Origin
3:56 minutesProblem 55
Textbook Question
Identify each equation without completing the square. 100x2 - 7y2 + 90y - 368 = 0
Verified step by step guidance1
Rewrite the given equation to group the terms involving \( y \) together: \( 100x^2 - 7y^2 + 90y - 368 = 0 \).
Focus on the quadratic terms: \( 100x^2 \) and \( -7y^2 \). Notice that the \( x^2 \) term is positive and the \( y^2 \) term is negative, which suggests the equation might represent a hyperbola.
Look at the linear term in \( y \), which is \( 90y \). This term shifts the center of the conic but does not change its fundamental type.
Recall the general forms of conic sections:
- Circle: \( Ax^2 + Ay^2 + ... = 0 \) with equal coefficients for \( x^2 \) and \( y^2 \)
- Ellipse: \( Ax^2 + By^2 + ... = 0 \) with \( A \) and \( B \) positive and unequal
- Hyperbola: \( Ax^2 - By^2 + ... = 0 \) or \( -Ax^2 + By^2 + ... = 0 \) with opposite signs for \( x^2 \) and \( y^2 \)
- Parabola: only one squared term
Based on the signs and coefficients of the squared terms, identify the conic as a hyperbola without completing the square.
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Types of Conic Sections
Conic sections are curves obtained by intersecting a plane with a cone, including circles, ellipses, parabolas, and hyperbolas. Each type has a distinct general equation form, and identifying the conic involves analyzing the coefficients of the squared terms and their signs.
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General Form of a Conic Equation
The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 represents conic sections. By examining coefficients A and C, and the presence or absence of the Bxy term, one can classify the conic without completing the square.
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Discriminant of a Conic Section
The discriminant, given by B² - 4AC, helps identify the conic type: if it is less than zero, the conic is an ellipse or circle; if zero, a parabola; and if greater than zero, a hyperbola. This method allows classification without rewriting the equation.
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