A conic (section) is the locus of a point moving in a plane such that its distance from a fixed point (focus) is in a constant ratio to its perpendicular distance from a fixed line (i.e. directrix). This constant ratio is called eccentricity of the conic.
The eccentricity of a circle is zero. It shows how “un-circular” a curve is. Higher the eccentricity, lower curved it is.
Axis of conic: Line passing through focus, perpendicular to the directrix.
Vertex: Point of the intersection of conic and axis.
Chord: Line segment joining any 2 points on the conic.
Double ordinate: Chord perpendicular to the axis
Latus Rectum: Double ordinate passing through focus.
|Standard Equation||Directrix||Focus||Length of Latus rectum||Vertex|
|y2 = 4ax||x = – a||S : (a, 0)||4a||(0, 0)|
|y2 = – 4ax||x = a||(- a, 0)||4a||(0, 0)|
|x2 = 4ay||y = – a||(0, +a)||4a||(0, 0)|
|x2 = – 4ay||y = a||(0, -a)||4a||(0, 0)|
Important results of a Parabola
1. 4 x distance between vertex and focus = Latus rectum = 4a.
2. 2 x Distance between directrix and focus = Latus rectum = 2(2a).
3. Point of intersection of Axis and directrix and the focus is bisected by the vertex.
4. Focus is the mid point of the Latus rectum.
5. (Distance of any point on parabola from axis)2 = (LR) (Distance of same point from tangent at vertex)
Focal Distance of a point
PS = PM = a + x1 = a + at2
Equation of chord for y2 = 4ax
Focal chord: Goes through (a, 0)
Length of focal chord:
PQ = PS + SQ
PQ = a(2cosecα)2 = 4acosec2α
It is a locus of a point which moves such that the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix) is always constant and less than 1, i.e o < e < 1.
Ellipse with Horizontal major axis
Focus : There are 2 focii; (ae, 0) and (-ae, 0)
Directrix: These foci have corresponding directrices as
xx’ : Major Axis: Length: 2a
yy’ : Minor axis; length : 2b
Vertox : (a, 0) & (-a, 0)
Centre : (0, 0)
Length of Latus rectum:
Ellipse with vertical major axis
Length of major axis : 2b
Length of minor axis : 2a
Focii : (0, be) & (0, -be)
Directrices: x = b/e & x = -b/e
Latus rectum: y = + be
Length of Latus rectum =
3. Distance between 2 directices:
4. Distance between 2 focii: (major axis) × eccentricity
5. Distance between focus and directrix:
If is the locus of a point which moves such that the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix) is always constant and greater than 1.
Focus: There are 2 focii (ae, 0) and (-ae, 0)
Directrix : The foci has corresponding directrices as x = +a/e and x = −a/e respectively.
Axis: xx’ : Transverse axis ; Length : 2a
yy’ : Conjugate axis ; Length : 2b (Hypothetical)
Vertex (0, 0) and (-a, 0)
Centre (0, 0)
Latus rectum: x = +ae
Length of latus rectum =
Position of point at hyperbola
If S1 > 0, point C lies inside the hyperbola
S1 = 0 point B lies on the hyperbola
S1 < 0 point A lies outside the hyperbola.
Ellipse and Hyperbola
Example 1: Find equation of a conic whose focus is at (1, 0) and directrix is 2x + 5y + 1 = 0. Also,
Squaring both sides, we get
4x2 + 25y2 + 1 + 20xy + 10y + 4x
Which is the required equation.
Example 2: If extreme points of LR are (11/2, 6) and (13/2, 4). Find the equation of the parabola.
Mid point of LR : focus : (6, 5)
Now, 4a = 2 or a = ½
The equation of parabolas are:
(y – 5)2 = 2(x – 5.5) and (y – 5)2 = – 2 (x – 6.5)