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Conic Section (Notes)

Conic Section (Notes)

Definition: The locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point to its distance from a fixed line is constant is called a conic section.

The fixed point is called focus, the fixed line is called directrix and the constant ratio is called the eccentricity (e). Any line through focus and perpendicular to directrix is called the axis of conic section.

Note: (1) When the eccentricity is equal to unity i.e. e = 1 then the conic section is called parabola.

          (2) When the eccentricity is less than unity i.e. e < 1 then the conic section is called ellipse.

         (3) When the eccentricity is greater than unity i.e. e > 1 then the conic section is called hyperbola.

Parabola:-

The locus of a point in a plain which moves in such a way that it is always equidistant from a fixed point (focus) and from a fixed line (directrix) is called a parabola.
In other words, the conic section whose eccentricity is equal to unity i.e. e = 1 is called a parabola.


Equation of Parabola (standard form):-

Let us place the parabola in standard position such that its vertex is at origin and axis along x-axis. Let A(0,0) be the vertex, SZ be the axix of the parabola. Let AS = AZ = a then the co-ordinate of focus(S) be (a,0). Consider a point P(x, y) on the parabola. From P, PM ⊥ QR is drawn also P & S are joined. Since the line QR at a distance -a from vertex is directrix, so the equation of directrix is x = -a or x + a = 0. Then PM = (x + a).
Now for the parabola, 
PM = PS
or, (PM)2 = (PS)2
or, (x + a)2 = (x - a)2 + (y - 0)2
or, x2. + 2ax + a2 = x2 - 2ax + y2
or, 4ax = y2
∴ y2 = 4ax is required equation of parabola in standard form
There are 4 types of standard equations of parabola, viz
y2 = 4ax,             y2 = -4ax,             x2 = 4ay,             x2 = -4ay

  1. y2 = 4ax :
    1. vertex - (0, 0)
    2. Focus - (a, 0)
    3. Equation of directrix is x = -a or x + a = 0
    4. Axis of parabola is x-axis i.e. y = 0
    5. Figure:- 
  2. y2 = -4ax :
    1. Vertex: (0, 0)
    2. Focus: (-a, 0)
    3. Equation of directrix is x = a or x - a = 0
    4. Axis of parabola is x-axis i.e. y = 0
    5. Figure:-

  3. x2 = 4ay
    1. Vertex: (0, 0)
    2. Focus: (0, a)
    3. Equation of directrix is y = -a or y + a = 0
    4. Axis of parabola is y-axis i.e. x = 0
    5. Figure:-

  4. x2 = -4ay:
    1. Vertex: (0, 0)
    2. Focus: (0, -a)
    3. Equation of directrix is y = a or y - a = 0
    4. Axis of parabola is y-axis i.e. x = 0
    5. Figure:-


Focal Distance:-

The distance of any point on the parabola from the focus is called focal distance (SP, SL, SQ, SL, etc.).

Focal Chord:-

Any chord of parabola passing throungh the focus is called focal chord (PQ, LL1).

Latus Rectum:-

The chord of parabola passing through focus and perpendicular to axis of parabola is called latus rectum. The length of latus rectum of parabola is 4a i.e. LL1 = 2a + 2a = 4a i.e. y = ±2a



Equation of parabola with its axis parallel to x-axis and vertex at any point (h, k):

Let A(h, k) be the vertex of parabola whose axis SZ is parallel to x-axis. Let, AS = AZ = a then the co-ordinate of focus S is (h + a, k) and the directrix QR is at distance 'a' from vertex A then equation of directrix QR is x = h - a or x - h + a = 0. Consider a point P(x,y) on the parabola the PM = x - (h - a).
Now, for parabola PM = PS,
or PM2 = PS2
or, {x - (h - a)}2 = {x - (h + a)}2 + (y - k)2
or, x2 - 2x(h - a) + (h - a)2 = x2 - 2x(h + a) + (h + a)2 + (y - k)2
or, -2hx + 2ax + h2 - 2ha + a2 = -2hx - 2ax + h2 + 2ah + a2 + (y - k)2
or, 4ax - 4ha = (y - k)2
or, (y - k)2 = 4a(x - h) is required equation of parabola.


Note: The equation of parabola whose vertex is at any point (h, k) and axis parallel to y-axis is (x - h)2 = 4a(y - k)


General equation of Parabola (Equation of parabola whose focus is at any point (h, k):

Let S(h, k) be the efocus and ax + by + c = 0 be the equation of directrix of the parabola. Consider a point P(x, y) on the parabola. A perpendicular PM is drawn to the directrix. Now, for parabola
PS = PM
or, PS2 = PM2  

Which is required equation of parabola in general form.



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