F. Conic Sections

The conic section in analytical geometry represents the curves that have been formed from curved lines, and have been defined with reference to a fixed point called the focus and the fixed-line called the directrix. The important conics are the circle, parabola, ellipse and the hyperbola. The standard form of equations of the different conics is as follows.

· Circle: x²+y²= a²

· Parabola: y²= 4ax when a>0

· Ellipse: x²/a² + y²/b² = 1

· Hyperbola: x²/a² – y²/b² = 1

Circle

The circle has a center and radius. A circle represents the locus of a point such that it's distance from a fixed point called the center is equal to a constant value called the radius. The general equation of a circle having the center at (h, k), and having a radius of r units is (x - h)² + (y - k)² = r². Further, the standard equation of a circle having the center at (0, 0), and the radius of 'a' units is x²+y²= a².

Conic equations

Parabola

A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed-line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola. The general equation of a parabola is: y = a(x-h)² + k or x = a(y-k)² +h, where (h,k) denotes the vertex. The standard equation of a regular parabola is y² = 4ax.

Ellipse

An ellipse in math is the locus of a plane point in such that its distance from a fixed point has a constant ratio 'e' to its distance from a fixed line, which is less than 1. The fixed point is called the focus and is denoted by S, the constant ratio e is the eccentricity, and the fixed line is called as directrix (d) of the ellipse. Also, an ellipse is the locus of a point, the sum of whose distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse. The standard equation of an ellipse is x² y²= 1

a²b²

Hyperbola

A hyperbola is a set of points whose difference of distances from two foci is a constant value. This difference is taken from the distance from the farther focus and then the distance from the nearer focus. For a point P(x, y). on the hyperbola and for two foci F, F', the locus of the

hyperbola is PF - PF' = 2a. The equation x² y²= 1

a²b² represents the standard form of the equation of a hyperbola. Here the x-axis is the transverse axis of the hyperbola, and the y-axis is the conjugate axis of the hyperbola.