Properties of parabola:
- A parabola is a set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
- The axis of symmetry of a parabola is a line that passes through the focus and is perpendicular to the directrix.
- The vertex of the parabola is the point where the axis of symmetry intersects the parabola.
- The standard form of a parabola with vertex at the origin is y^2 = 4px, where p is the distance from the vertex to the focus.
Properties of ellipse:
- An ellipse is a set of all points such that the sum of the distances from two fixed points (called foci) is constant.
- The major axis of an ellipse is the longest diameter and contains the two foci. The minor axis is the perpendicular diameter.
- The center of an ellipse is the midpoint of the major axis.
- The standard form of an ellipse with the center at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the semi-major and semi-minor axes.
Properties of hyperbola:
- A hyperbola is a set of all points such that the difference of the distances from two fixed points (called foci) is constant.
- The transverse axis of a hyperbola is the longest diameter and contains the two vertices. The conjugate axis is the perpendicular diameter.
- The center of a hyperbola is the midpoint of the transverse axis.
- The standard form of a hyperbola with the center at the origin is x^2/a^2 - y^2/b^2 = 1, where a and b are the lengths of the semi-transverse and semi-conjugate axes.
Important Points to Remember in these Properties of parabola, ellipse and hyperbola:
- A parabola is the set of all points that are equidistant from a fixed point (focus) and a fixed line (directrix).
- An ellipse is the set of all points such that the sum of the distances from two fixed points (foci) is constant.
- A hyperbola is the set of all points such that the difference of the distances from two fixed points (foci) is constant.
Standard forms with vertex at origin:
- The standard form of a parabola with vertex at the origin is y^2 = 4px, where p is the distance from the vertex to the focus.
- The standard form of an ellipse with the center at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the semi-major and semi-minor axes.
- The standard form of a hyperbola with the center at the origin is x^2/a^2 - y^2/b^2 = 1, where a and b are the lengths of the semi-transverse and semi-conjugate axes.
- For all three types of conic sections, if the vertex is not at the origin, the equations can be shifted by replacing x with (x - h) and y with (y - k), where (h, k) is the coordinates of the vertex.
Important Formulas to Remember in these Standard forms with vertex at origin:
- Parabola: $$y^2 = 4px$$
- Ellipse: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
- Hyperbola: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
Additional properties of conic sections:
- The eccentricity of an ellipse is the ratio of the distance between the foci to the length of the major axis. It is always less than 1.
- The eccentricity of a hyperbola is the ratio of the distance between the foci to the length of the transverse axis. It is always greater than 1.
- The directrix of a parabola is a line that is equidistant to the focus and perpendicular to the axis of symmetry.
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The asymptotes of a hyp-erbola are two lines that the hyperbola
approaches but never intersects. The slopes of the asymptotes are ±(b/a).