Parabola
Conic Section
A conic is the locus of a point whose distance from a fixed point bears a
constant ratio to its distance from a fixed line. The fixed point is the
focus S and the fixed line is directrix l.
Z
P
M
directrix
S
(focus)
Z′
The constant ratio is called the eccentricity denoted by e.
(i) If 0 < e < 1, conic is an ellipse.
(ii) e = 1, conic is a parabola.
(iii) e > 1, conic is a hyperbola.
General Equation of Conic
If fixed point of curve is ( x1 , y1 ) and fixed line is ax + by + c = 0, then
equation of the conic is
( a 2 + b2 ) [( x − x1 )2 + ( y − y1 )2 ] = e2( ax + by + c)2
which on simplification takes the form
ax 2 + 2hxy + by 2 + 2gx + 2 fy + c = 0,
where a , b, c, f , g and h are constants.
A second degree equation ax 2 + 2hxy + by 2 + 2gx + 2 fy + c = 0
represents
a h g
(i) a pair of straight lines, if ∆ = h b f = 0
g f c
, (ii) a pair of parallel (or coincident) straight lines, if ∆ = 0
and h 2 = ab.
(iii) a pair of perpendicular straight lines, if ∆ = 0 and a + b = 0
(iv) a point, if ∆ = 0 and h 2 < ab
(v) a circle, if a = b ≠ 0, h = 0 and ∆ ≠ 0
(vi) a parabola, if h 2 = ab and ∆ ≠ 0
(vii) a ellipse, if h 2 < ab and ∆ ≠ 0
(viii) a hyperbola, if h 2 > ab and ∆ ≠ 0
(ix) a rectangular hyperbola, if h 2 > ab, a + b = 0 and ∆ ≠ 0
Important Terms Related to Parabola
(i) Axis A line perpendicular to the directrix and passes through
the focus.
(ii) Vertex The intersection point of the conic and axis.
(iii) Centre The point which bisects every chord of the conic passing
through it.
(iv) Focal Chord Any chord passing through the focus.
(v) Double Ordinate A chord perpendicular to the axis of a conic.
(vi) Latusrectum A double ordinate passing through the focus of
the parabola.
(vii) Focal Distance The distance of a point P ( x , y ) from the focus S
is called the focal distance of the point P.
Parabola
A parabola is the locus of a point which moves in a plane such that its
distance from a fixed point in the plane is always equal to its distance
from a fixed straight line in the same plane.
If focus of a parabola is S ( x1 , y1 ) and equation of the directrix is
ax + by + c = 0, then the equation of the parabola is
( a 2 + b2 )[( x − x1 )2 + ( y − y1 )2 ] = ( ax + by + c)2
Y
, y)
P(x
X' X
O S(x1, y1)
ax + by + c = 0
Y'
Conic Section
A conic is the locus of a point whose distance from a fixed point bears a
constant ratio to its distance from a fixed line. The fixed point is the
focus S and the fixed line is directrix l.
Z
P
M
directrix
S
(focus)
Z′
The constant ratio is called the eccentricity denoted by e.
(i) If 0 < e < 1, conic is an ellipse.
(ii) e = 1, conic is a parabola.
(iii) e > 1, conic is a hyperbola.
General Equation of Conic
If fixed point of curve is ( x1 , y1 ) and fixed line is ax + by + c = 0, then
equation of the conic is
( a 2 + b2 ) [( x − x1 )2 + ( y − y1 )2 ] = e2( ax + by + c)2
which on simplification takes the form
ax 2 + 2hxy + by 2 + 2gx + 2 fy + c = 0,
where a , b, c, f , g and h are constants.
A second degree equation ax 2 + 2hxy + by 2 + 2gx + 2 fy + c = 0
represents
a h g
(i) a pair of straight lines, if ∆ = h b f = 0
g f c
, (ii) a pair of parallel (or coincident) straight lines, if ∆ = 0
and h 2 = ab.
(iii) a pair of perpendicular straight lines, if ∆ = 0 and a + b = 0
(iv) a point, if ∆ = 0 and h 2 < ab
(v) a circle, if a = b ≠ 0, h = 0 and ∆ ≠ 0
(vi) a parabola, if h 2 = ab and ∆ ≠ 0
(vii) a ellipse, if h 2 < ab and ∆ ≠ 0
(viii) a hyperbola, if h 2 > ab and ∆ ≠ 0
(ix) a rectangular hyperbola, if h 2 > ab, a + b = 0 and ∆ ≠ 0
Important Terms Related to Parabola
(i) Axis A line perpendicular to the directrix and passes through
the focus.
(ii) Vertex The intersection point of the conic and axis.
(iii) Centre The point which bisects every chord of the conic passing
through it.
(iv) Focal Chord Any chord passing through the focus.
(v) Double Ordinate A chord perpendicular to the axis of a conic.
(vi) Latusrectum A double ordinate passing through the focus of
the parabola.
(vii) Focal Distance The distance of a point P ( x , y ) from the focus S
is called the focal distance of the point P.
Parabola
A parabola is the locus of a point which moves in a plane such that its
distance from a fixed point in the plane is always equal to its distance
from a fixed straight line in the same plane.
If focus of a parabola is S ( x1 , y1 ) and equation of the directrix is
ax + by + c = 0, then the equation of the parabola is
( a 2 + b2 )[( x − x1 )2 + ( y − y1 )2 ] = ( ax + by + c)2
Y
, y)
P(x
X' X
O S(x1, y1)
ax + by + c = 0
Y'
