**DEFINITION :**

__CONIC SECTION__

**A conic section or conic is the locus of a point, which moves in a plane so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line, not passing through the fixed point.**

∎ The fixed point is called the **focus.**

∎ The fixed straight line is called the **directrix.**

∎ The constant ratio is called the **eccentricity** and is denoted by e.

∎ **When the eccentricity is unity; i.e., e = 1 , the conic is called a parabola; **

**when e < 1, the conic is called an ellipse .**

**and when e > 1, the conic is called a hyperbola.**

∎ The straight line passing through the focus and perpendicular to the directrix is called the **axis of the parabola.**

∎ A point of intersection of a conic with its axis is called **vertex.**

__Standard equation of a Parabola:__

Let S be the focus, ZM the directrix and P the moving point. Draw SZ perpendicular from S on the directrix. Then SZ is the axis of the parabola. Now the middle point of SZ , say A , will lie on the locus of P ,

i.e., AS = AZ

Take A as the origin, the x-axis along AS, and the y-axis along the perpendicular to AS at A, as in the figure.

Let AS = a , so that ZA is also a .

Let (x, y) be the coordinates of the moving point P.

Then MP = ZN = ZA + AN = a + x .

But by definition MP = PS

⇒ MP^{2} = PS^{2}

So that , (a + x)^{2} = (x – a)^{2} + y^{2}

Hence , the equation of parabola is y^{2} = 4ax

__Latus Rectum:__

The chord of a parabola through the focus and perpendicular to the axis is called the latus rectum.

In the figure LSL’ is the latus rectum.

Also LSL’ = 4a = double ordinate through the focus S.

**Note:**

∎ Any chord of the parabola y^{2} = 4ax perpendicular to the axis of the parabola is called **double ordinate.**

∎ Two parabolas are said to be equal when their **latus recta are equal.**