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
⇒ MP2 = PS2
So that , (a + x)2 = (x – a)2 + y2
Hence , the equation of parabola is y2 = 4ax
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.
∎ Any chord of the parabola y2 = 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.