How can you distinguish parabolas from other conic sections by looking at their equations?
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To distinguish parabolas from other conic sections by looking at their equations, you can focus on the highest power of the variable present in the equation.
A parabola is defined by an equation in the form: y = ax² + bx + c or x = ay² + by + c. Here, the highest power of the variable (x or y) is 2. The coefficients a, b, and c can also give you additional information about the orientation, vertex, and other properties of the parabola.
On the other hand, other conic sections such as circles, ellipses, and hyperbolas have equations where the highest power of the variable is either 1 (linear) or 0 (constant). Their general equations are:
- Circle: (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is the radius.
- Ellipse: (x - h)²/a² + (y - k)²/b² = 1 or (x - h)²/b² + (y - k)²/a² = 1, where (h, k) represents the center of the ellipse and a and b are the semi-major and semi-minor axes, respectively.
- Hyperbola: (x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/a² - (x - h)²/b² = 1, where (h, k) represents the center of the hyperbola, a and b are the distances from the center to the vertices, and a² and b² give you information about the hyperbola's shape.
By examining the highest power of the variable in an equation, you can determine if it represents a parabola or one of the other conic sections.
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