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Magazine Chapter 10: Problem 51
Rewrite each equation in one of the standard forms of the conic sections and identify the conic section. $$25 x^{2}-150 x-8 y=4 y^{2}-121$$
Short Answer
Expert verified
The given equation represents a hyperbola.
Step by step solution
01
- Rearrange the equation
Move all the terms to one side to set the equation to zero. Starting from the equation: \[25 x^{2}-150 x-8 y = 4 y^{2}-121\] Rearrange it to \[25 x^{2}-150 x - 8 y - 4 y^{2} + 121 = 0\]
02
- Group the terms
Group the x and y terms together to facilitate completing the square: \[25 x^{2} - 150 x - 4 y^{2} - 8 y = -121\]
03
- Complete the square for x
Factor out the coefficient of x terms: \[25(x^{2}-6x) - 4 y^{2} - 8 y = -121\] Complete the square for x: \[25((x-3)^{2} - 9) - 4 y^{2} - 8 y = -121\] Distribute and simplify: \[25(x-3)^{2} - 225 - 4 y^{2} - 8 y = -121\]
04
- Complete the square for y
Factor out the coefficient of y terms: \[25(x-3)^{2} - 225 - 4(y^2 + 2y + 1 -1) = -121\] Complete the square for y: \[25(x-3)^{2} - 4(y+1)^{2} = -121 + 225 + 4\]
05
- Simplify and rewrite in standard form
Combine the constants on the right side: \[25(x-3)^{2} - 4(y+1)^{2} = 108\] Divide both sides by 108 to rewrite the equation in standard form: \[ \frac{(x-3)^{2}}{4.32} - \frac{(y+1)^{2}}{27} = 1\]
06
- Identify the conic section
The equation \( \frac{(x-3)^{2}}{4.32} - \frac{(y+1)^{2}}{27} = 1\) is in the standard form of a hyperbola. Hence, the conic section is a hyperbola.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a technique used in algebra to simplify quadratic expressions. It involves creating a perfect square trinomial from a quadratic term and a linear term. This helps in rewriting quadratic equations into a standard form, making them easier to solve or graph.
To complete the square, follow these steps:
To complete the square, follow these steps:
- Identify the quadratic, linear, and constant terms in the expression. For example, in the expression \(ax^2 + bx + c\), the quadratic term is \(ax^2\) and the linear term is \(bx\).
- Factor out the coefficient of the quadratic term from both the quadratic and linear terms if it's not 1. For example, if you have \(25x^2 - 150x\), factor out 25 to get \(25(x^2 - 6x)\).
- Add and subtract the square of half the coefficient of the linear term inside the parentheses. For the expression \(x^2 - 6x\), take half of -6 (which is -3), square it to get 9, and then write \((x - 3)^2 - 9\). The expression then becomes \(25((x-3)^2 - 9)\).
- Distribute and simplify as required.
Standard Form of Conic Sections
Conic sections are curves obtained by intersecting a cone with a plane. They are classified into four types: circles, ellipses, parabolas, and hyperbolas. Every conic section has a standard form equation that makes them easier to identify and graph.
The standard forms are as follows:
This helps you quickly identify the type of conic section and analyze its properties.
The standard forms are as follows:
- Circle: \((x - h)^2 + (y - k)^2 = r^2\)
- Ellipse: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\)
- Hyperbola: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\)
- Parabola: Horizontal: \(y = a(x - h)^2 + k\), Vertical: \(x = a(y - k)^2 + h\)
This helps you quickly identify the type of conic section and analyze its properties.
Hyperbola Identification
A hyperbola is a type of conic section defined by its distinct shape consisting of two disconnected curves. It can be easily identified if its standard form equation adheres to the structure: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).
Key features of hyperbolas include:
Key features of hyperbolas include:
- A center at \((h, k)\)
- Two vertices located \(a\) units away from the center along the transverse axis
- Two foci located \(c = \sqrt{a^2 + b^2}\) units from the center
- Asymptotes that guide the shape of the hyperbola, intersecting at the center
- Ensure the coefficients of \(x^2\) and \(y^2\) have opposite signs.
- Rewrite the equation into the standard form by completing the square and simplifying.
- Check the structure against the standard form to confirm it's a hyperbola.
