Conic section definition the best Let L be a **fixed line in a plane** and **F be a fixed point** not on the line L. suppose that **|PM|** denote the distance of the point **p(x, y)** from the line L. the set of all points Pin the plane such that** |PF| ÷|PM|=** e (positive constant) is **called a conic section.**

**Conic section definition types best**

**(i) If e= 1, then the conic is parabola **

**(ii) If 0 < e < 1 then the conic is an ellipse.**

**(iii) If e > 1 then the conic is hyperbola.**

The fixed line is called the directrix and the fixed point F is called a focus

of the conic. The number e is called the eccentricity of the conic.

**General form of an equation of a parabola:**

Let F(h, k) be the focus of the line, l x + my + n=0 be the DirectX of the parabola of a parabola. An equation of the parabola can be derived by the definition of the parabola.

Let P(x, y) be a point of the parabola. The length of the perpendicular PM from p(x, y) to the Directix is given by

that is

is a general form of an equation of the parabola.

**Second degree equation of the form of parabola:**

A second degree equation of the form

**ax² + by²+2gx+ 2fy+ c=0**

with either a= 0 or b =0 but not both zero, to represent a parabola the equation can be

analyzed by completing the square

**Parabola types: **

(1) if the focus lies on the y-axis with coordinate F(0, a) and directrix of the parabola

y= -a then equation of the parabola is

x² = 4ay

(2) if the focus is F(0,a) and directrix is the line y = a then the equation of the parabola is

x² = -4ay

(3) If the focus of the parabola is F (-a, 0) and its is the line x = a then the equation of the

parabola is

y² = -4ax

the curve is symmetric with respect to the x-axis and lies in the second and third quadrant.

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