Parabolas, Ellipses and Hyperbolas (AS)

General Definition of Conic Sections: A conic section is the intersection of a plane and a cone (either a single or a double cone). Depending on the angle of intersection, the shape may be a parabola, ellipse or hyperbola.
Parabolas: A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard form of a parabola is y = ax²+bx+c or x = ay²+by+c.
Deriving the Vertex Form of a Parabola Equation: If a parabola in standard form is y = ax² + bx + c, then the vertex form is y = a(x - h)² + k, where the vertex (h, k) can be found using -b/2a and substituting this back into the equation.
Ellipses: An ellipse is a curve where the sum of the distances from any point on the curve to two fixed points (the foci) is a constant. The standard form of an ellipse with a horizontal major axis is (x-h)²/a² + (y-k)²/b² = 1 and for a vertical major axis it is (x-h)²/b² + (y-k)²/a² = 1.
Major and Minor Axes of Ellipses: In an ellipse, the major axis is the longest diameter and the minor axis is the shortest diameter. The foci lie along the major axis, equidistant from the centre.
Hyperbolas: A hyperbola is a set of all points in a plane where the difference of the distances from any point on the curve to two fixed points (the foci) is a constant. The standard form of a hyperbola with a horizontal transverse axis is (x-h)²/a² - (y-k)²/b² = 1 and for a vertical transverse axis it is (x-h)²/b² - (y-k)²/a² = 1.
Asymptotes of Hyperbolas: Asymptotes of a hyperbola are the lines that the hyperbola approaches as it extends to infinity. The equations of these lines can be found by changing the equation of the hyperbola to be equal to zero.
Eccentricity: It’s a measure of how much a conic section deviates from being circular. For an ellipse, it’s the ratio of the distance between the foci to the length of the major axis; for a hyperbola, it’s the ratio of the distance between the foci to the length of the average of real-asymptote-intercepts.

Refer to past papers and work through examples of each key point identified above in order to thoroughly understand these topics. Remember, practise and revision are the keys to success in Further Pure Mathematics 1.