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  Concepts of Conics and Parametric Equations

Concepts of Conics and Parametric Equations:

Chapter Summary:

Conics:

The conics get their name from the fact that they can be made by passing a plane via a double-napped cone. There are mainly four conic sections, and three degenerate cases, though, in this class we are going to look at five degenerate cases which can be formed from the general second degree equation.

The general form of second degree equation is given by the Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Parabola:

The parabola is ‘the set of all points in a plane equidistant from a fixed point (that is, focus) and a fixed line (or directrix)’.

The distances to any point (x, y) on parabola from the focus (0, p) and the directrix y = -p, are equal to one other. This can be employed to develop the equation of the parabola.

Circle:

The circle is ‘the set of all points in a plane equidistant from the fixed point (that is, center)’.

The standard form for a circle, with center at origin is x2 + y2 = r2, here r is the radius of the circle.

Ellipse:

The ellipse is ‘the set of all points in a plane in such a way that the sum of distances from the two fixed points (that is, foci) is constant’.

The sum of distances to any point on the ellipse (x, y) from the two foci (c,0) and (-c,0) is constant. That constant will be equal to 2a.

When we let d1 and d2 bet the distances from foci to the point, then d1 + d2 = 2a.

Key Topics included in this chapter are as follows:

a) Conics
b) Translations of Conics
c) Parametric Equations

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