Given: r = 4/-2-costheta
What type of conic does this represent?
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Circle
Ellipse
Hyperbola
Parabola
To determine the type of conic section represented by the equation r = 4/-2cos(theta), we need to transform it into a standard form equation for each conic section and analyze the resulting equation.
Let's go through each option and see which one applies.
1. Circle:
The standard form equation for a circle in polar coordinates is r = a, where 'a' is a constant representing the radius. The given equation, r = 4/-2cos(theta), does not match the standard form equation for a circle as it includes the cos(theta) term. Therefore, the equation does not represent a circle.
2. Ellipse:
The standard form equation for an ellipse in polar coordinates is r = e / (1 + εcos(theta)), where 'e' represents the distance from the center to either focus, and 'ε' represents the eccentricity. The given equation, r = 4/-2cos(theta), does not match the standard form equation for an ellipse as it does not have the (1 + εcos(theta)) term. Therefore, the equation does not represent an ellipse.
3. Hyperbola:
The standard form equation for a hyperbola in polar coordinates is r = e / (εcos(theta)), where 'e' represents the distance from the center to either focus, and 'ε' represents the eccentricity. The given equation, r = 4/-2cos(theta), matches the standard form equation for a hyperbola as it has the εcos(theta) term. Therefore, the equation represents a hyperbola.
4. Parabola:
The standard form equation for a parabola in polar coordinates is r = a / (1 + εcos(theta)), where 'a' represents the focal length of the parabola, and 'ε' represents the eccentricity. The given equation, r = 4/-2cos(theta), does not match the standard form equation for a parabola as it does not have the (1 + εcos(theta)) term. Therefore, the equation does not represent a parabola.
Based on the analysis, the equation r = 4/-2cos(theta) represents a hyperbola.