Questions and Answers
intersecting plane to a double-napped cone?
Conic Section
eccentricity of all the conic section?
circle, e=0, parabola, e=1 ellipse, e<1, hyperbola, e>1.
what is the general form of equation?
Ax²+Bxy+Cy²+Dx+Ey+F=0
what is the quadratic terms of all the conic section?
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what is the discriminat formula?
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what type of conic section is B²-4AC<0?
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what type of conic section is B²-4AC=0?
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what conic section is B²-4AC>0?
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How do u identify if a discriminant value is a ellipse or circle?
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what is a set of point?
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what is a directorix
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special points with reference to which any form of variety of curves is directed
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what is eccentricity
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Study Notes
Intersecting Planes and Conic Sections
- A conic section is formed by the intersection of a double-napped cone and a plane.
- The angle and position of the intersecting plane determine the type of conic section produced (circle, ellipse, parabola, or hyperbola).
Eccentricity of Conic Sections
- The eccentricity (e) measures how much a conic section deviates from being circular.
- Eccentricity values:
- e = 0 corresponds to a circle.
- 0 < e < 1 corresponds to an ellipse.
- e = 1 corresponds to a parabola.
- e > 1 corresponds to a hyperbola.
General Form of a Conic Section
- The general quadratic equation of conic sections is given by:
- Ax² + Bxy + Cy² + Dx + Ey + F = 0.
- A, B, C, D, E, and F are constants.
Quadratic Terms of Conic Sections
- The quadratic terms refer to Ax², Bxy, and Cy² in the general equation.
- The coefficients A, B, and C play crucial roles in identifying the type of conic section represented.
Discriminant Formula
- The discriminant for conic sections is defined as:
- D = B² - 4AC.
- This formula helps classify the conic section based on the values of A, B, and C.
Classification of Conic Sections Using Discriminant
- B² - 4AC < 0 indicates an ellipse or a circle.
- B² - 4AC = 0 indicates a parabola.
- B² - 4AC > 0 indicates a hyperbola.
Identifying Ellipses and Circles
- To determine whether a conic section is an ellipse or circle:
- Check the coefficients A and C:
- If A = C and B = 0, the conic is a circle.
- If A ≠ C and B = 0, the conic is an ellipse.
- Check the coefficients A and C:
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