Definition:
Based on a fixed point (S) and a fixed line ( l ) there exists a particle P which is moving in a plane such that
Then the locus of P is called a conic.
Fixed point (S) is the focus, fixed line ( l ) is the directrix and e is called the eccentricity. According to 'e' conics are of three types
(1) ⇒ If e = 1 then the conic is called a Parabola
(2) ⇒ If e < 1 then the conic is called an Ellipse
(3) ⇒ If e > 1 then the conic is called a Hyperbola
Equation of conic :
Definition:
A Parabola is the set of points in a plane, each of which is equidistant from the focus and the directrix.
Equation of parabola
⇒ PS = PM
Conceptual Theorem
1. Show that the equation of the parabola in the standard form is y2 = 4ax.
Proof :
Let 'S' be the focus and the line "l" be the directrix for the parabola
Let 'A' be the mid point of SZ and SA = AZ = a
Choose and as X and Y - axes
then A = (0, 0), S = (a, 0) and Z = (-a, 0)
Let P (x, y) be any point on the parabola
Draw PM
PM = ZN = ZA + AN = x + a
From the definition of Parabola: PS = PM
This is the standard form of the parabola.
Now we define some important words which play a vital role in this chapter
Axis : The line which is perpendicular to directrix and which passes through focus.
Focal chord : The chord which passes through focus.
Lotus rectum : The focal chord which is perpendicular to the axis.
Vertex : The point of intersection of axis and conic.
Focal distance : The distance between focus and any point on the conic.
Different forms of parabola
1. Parabola: y2 = 4ax
Focus: S (a, 0)
Vertex: A (0, 0)
Directrix: x + a = 0
Equation of Latus rectum: x - a = 0
L.L.R.: 4a
Focal Distance:
Tangent at vertex: x = 0
2. Parabola: y2 = - 4ax
Focus: S(-a, 0)
Vertex: A (0, 0)
Directrix: x - a = 0
Axis : y = 0
Equation of Latus rectum: x + a = 0
L.L.R. = 4a
Focal distance:
Tangent at vertex: x = 0
3. Parabola: x2 = 4ay
Focus: S (0, a)
Vertex: A(0, 0)
Directrix: y + a = 0
Axis: x = 0
Equation of Latus rectum: y - a = 0
L.L.R: 4a
Focal distance =
Tangent at vertex: y = 0
4. Parabola: x2 = - 4ay
Focus: S (0, -a)
Vertex: A (0, 0)
Directrix: y - a = 0
Axis: x = 0
Equation of latus rectum: y + a = 0
Focal distance :
Tangent at vertex: y = 0
Focus: S (a + α, β)
Vertex: A (α, β)
Directrix: x + a - α = 0
Axis: y - β = 0
Equation of Latus rectum: x - a - α = 0
L.L.R: 4a
Focal distance
Tangent at vertex: x - α = 0
6. Parabola: (x - α)2 = 4a (y - β)
Focus: S(α, a + β)
Vertex: A(α, β)
Directrix: y + a - β = 0
Axis: x - α = 0
Equation of Latus rectum = y - a - β = 0
L.L.R: 4a
Focal distance:
Tangent at vertex: y - β = 0