A conic section is a section cut off from a right circular cone by a plane in various ways. The shape of the section depends upon the position of the cutting plane.
Section of a right circular cone by a plane parallel to its base is a circle as shown in the figure. When the plane passes through the vertex, the section is a point circle.
Section of a right circular cone by a plane not parallel to any generator of the cone and not perpendicular or parallel, to the axis of the cone is an ellipse.
Section of right circular cone by a plane parallel to a generator of the cone is a parabola
Section of a right circular cone by a plane parallel to the axis of the cone is a hyperbola
Section of a right circular cone by a plane passing through its vertex is a pair of straight lines passing through the vertex of the cone. When this plane touches the cone, the section is a line (coincident lines). And when this plane is not meeting any other part of the cone, the section is point.
Analytically, a conic section or conic is the locus of a point, which moves 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.Circle (e = 0) and pair of straight lines (e
) are degenerate cases of an ellipse and a hyperbola respectively.Circle
Definition:
A circle is the locus of a point which moves in such a way that its distance from a fixed point is constant. The fixed point is called the centre of the circle and the constant distance, the radius of the circle.
Equation of the Circle in Various Forms:
(i) The equation (x - a)2 + (y - b)2 = r2 represents a circle with centre (a, b) and radius r where, r =
(ii) The simplest equation of the circle is x2 + y2 = r2 whose centre is (0, 0) and radius `r'.
(iii) The equation x2 + y2 + 2gx + 2fy + c = 0 is the general equation of a circle with centre (-g, -f) and radius
.(iv) Equation of the circle with points P(x1, y1) and Q(x2, y2) as extremities of a diameter is (x - x1) (x - x2) + (y - y1) (y - y2) = 0.
Equation of a Circle under Different Conditions:
| Condition | Equation | |
(i) Touches the x-axis only with centre ( , a) and radius a | (x-)2 + (y-a)2 = a2 or x2 + y2- 2x - 2ay + 2 = 0 | |
(ii) Touches the y-axis only with centre (a,  ) and radius a | (x-a)2 + (y-)2 = a2 or x2+ y2 - 2ax - 2y + 2 = 0 | ||
| (iii) Touches both the axes with centre (a, a) and radius a | (x-a)2 + (y-a)2 = a2 or x2 + y2 - 2ax - 2ay + a2 = 0 | |
 | ||
(iv) Passes through the origin with centre
and radius 
x2 +y2 - x - y = 0Parametric Equations of a Circle:
The equations x = a cos
, y = a sin are called parametric equations of the circle x2 + y2 = a2 and is called a parameter. The point (a cos , a sin ) is also referred to as point . The parametric coordinates of any point on the circle (x - h)2 + (y - k)2 = a2 are (h + a cos, k + a sin) with 0 
< 2
. 
The Position of a Point with respect to a Circle:
The point P(x1, y1) lies outside, on, or inside a circle S
x2 + y2 + 2gx + 2fy + c = 0, according as S1 x12 + y12 + 2gx1 + 2fy1 + c >, = or < 0.The centre of the circle is C (- g, - f). The point P (x1, y1) lies outside the circle if CP is greater than the radius of the circle
The point P (x1, y1) lies inside the circle if CP < radius
Concentric circles
Two circles having the same centre but different radii r1 and r2 are called concentric circles. Thus (x - h)2 + (y - k)2 =
and (x - h)2 + (y - k)2 = 
, (r1 
r2) represent two concentric circles. Cyclic quadrilateral
If all the four vertices of a quadrilateral lie on a circle, then the quadrilateral is called a cyclic quadrilateral. The four vertices are said to be concyclic.
Parabola
Standard Equation
A parabola is the locus of a point whose distance from a fixed point S (called the focus) is equal to its distance from a fixed line ZM (called the directrix).
The straight line passing through the focus and perpendicular to the directrix is called the axis of the parabola.
The equation of parabola is y2 = 4ax. The point A (0, 0) is called the vertex of the parabola. The focus S is at (a, 0) and the equation of the directrix is x + a = 0.
Latus Rectum:
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'= 2
= double ordinate through the focus S. The coordinates of L are (a, 2a) and that of L' (a, - 2a).
Note:
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.Since y2 = 4ax is defined only for x 
0, the curve lies only on the right side of the y-axis.A line joining any two points on the curve is called a chord.Four common forms of a Parabola:
Form: y2 = 4ax y2 = - 4ax x2 = 4ay x2 = - 4ay
Vertex : (0, 0) (0, 0) (0,0) (0, 0)
Focus : (a, 0) (-a, 0) (0, a) (0, -a)
Equation of the Directrix: x = -a x = a y = -a y = a
Equation of the axis: y = 0 y = 0 x = 0 x = 0
Parametric Coordinates:
Any point on the parabola y2 = 4ax is (at2, 2at) and we refer to it as the point 't'. Here, t is a parameter, i.e., it varies from point to point.
Focal Distance of a Point:
The focal distance of any point P(x, y) on the parabola y2 = 4ax is the distance between the point P and the focus S, i.e. PS.
Thus the focal distance = PS = PM =ZN = ZA + AN = a + x.
Position of a point relative to the Parabola:
Consider the parabola y2 = 4ax. If (x1, y1) is a given point and y21 - 4ax1 = 0, then the point lies on the parabola. But when y12 - 4ax1
0, we draw the ordinatePM meeting the curve in L. Then P will lie outside the parabola if
PM > LM, i.e., PM2 - LM2 > 0.
The condition for P to lie inside the parabola is y12 - 4ax1 < 0.
Ellipse
Definition:
An ellipse is the locus of a point which moves such that its distance from a fixed point (focus) bears to its perpendicular distance from a fixed straight line (directrix) a constant ratio (eccentricity) which is less than unity.
Standard Equation:
+ 
= 1 or
, where b2 = a2(1 - e2).The eccentricity of the ellipse
is given by the relation b2 = a2(1 - e2), i.e., e2 = 1 - b2/a2.
Central Curve:
A curve is said to be a central curve if there is a point, called the centre, such that every chord passing through it is bisected at it.
Latus Rectum:
The chord LS'L' through the focus at right angles to the major axis is called the latus rectum.
The coordinates of L are
and those of L' are
. The length of the latus rectum is LL' =
. The equation of the latus rectum is x = ae.Note:
The major axis AA' is of length 2a and the minor axis BB' is of length 2b.The foci are (-ae, 0) and (ae, 0).The equations of the directrices are x = a/e and x = -a/e. The length of the semi latus rectum = b2 / a.Circle is a particular case of an ellipse with e = 0.C (0, 0) is called the centre of the ellipse.Focal Distance of a Point:
The distance of any point P on the curve from its focus is called its focal distance (or focal radius).
Since S'P = ePN' , SP = ePN,
S'P + SP = e(PN + PN')= e (NN') = e(2a/e) = 2a
the sum of the focal distances of any point on the ellipse is equal to its major axis.
Another Definition of Ellipse:
Ellipse is the locus of a point the sum of whose distances from two fixed points is a constant greater than the distance between the fixed points.
Other Forms:
If in the equation
, a2 < b2, then the major and minor axis of the ellipse lie along the y and the x-axes and are of lengths 2b and 2a respectively. Thefoci become (0,
be), and the directrices become y = b/e where e =
. The length of the semi-latus rectum becomes 
. If the centre of the ellipse be taken at (h, k) and axes parallel to the x and the y-axes, then the equation of the ellipse is
.Let the equation of the directrix of an ellipse be ax + by + c = 0 and the focus be (h, k). Let the eccentricity of the ellipse be e(e < 1).
If P(x, y) is any point on the ellipse, then
PS2 = e2 PM2
(x - h)2 + (y - k)2 = e2
, which is of the form ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ... (*) where
= abc +2 fgh -af2-bg2 - ch2 0, h2 < ab.
These are the necessary and sufficient conditions for a general quadratic equation given by (*) to represent an ellipse.
Position of a Point Relative to an Ellipse:
The point P(x1, y1) is outside or inside the ellipse
according as the quantity 
is positive or negative.Let P0 (x1, y0) be the corresponding point on the ellipse so that
If P (x1, y1), is lying outside the ellipse then y1 > y0
or 
.Parametric Equations of an Ellipse:
Clearly, x = a cos
, y = b sin satisfy the equation 
for all real values of .Hence, the parametric equations of the ellipse
are x = a cos, y = b sin where
is the parameter.Also (a cos
, b sin) is a point on the ellipse 
= 1 for all values of (0 < 2).The point (a cos
, b sin) is also called the point . Angle is called the eccentric angle of the point (a cos, b sin) on the ellipse.HYPERBOLA
Definition:
A hyperbola is the locus of a point which moves so that its distance from a fixed point (focus) bears to its distance from a fixed straight line (directrix) a constant
ratio (eccentricity) which is greater than unity.
Standard Equation:
The condition PS2 = e2. (distance of P from ZM)2 gives (x - ae)2 + y2 = e2 (x - a/e)2 or x2(1 - e)2 + y2 = a2(1 - e2)
i.e.
. .... (i) 
Since e>1, e2-1 is positive. Let a2(e2 -1) = b2. Then the equation (i) becomes
.The eccentricity e of the hyperbola
is given by the relation 
.Note:
The points A and A' where the straight line joining the two foci cuts the hyperbola are called the vertices of the hyperbola. The coordinates of A and A' are (a, 0) and (- a, 0) respectively.The straight line joining the vertices is called the transverse axis of the hyperbola, its length AA' is 2a.The middle point C of AA' possesses the property that it bisects every chord of the hyperbola passing through it. The straight line through the centre of a hyperbola which is perpendicular to the transverse axis does not meet the hyperbola in real points. If B and B' be the points on this line such that BC= CB' = b, the line BB' is called the conjugate axis.When the lengths of the transverse axis and the conjugate axis are equal i.e. a = b, then the equation of the hyperbola becomes x2 - y2 = a2. This is called a rectangular or equilateral hyperbola and in this case e2 = 1 + 1 i.e. eccentricity of a rectangular hyperbola is
.A latus rectum is the chord through a focus at right angle to the transverse axis.The length of the semi - latus rectum obtained by putting x = ae in the equation of the hyperbola is 
.A circle having its centre at the centre of the hyperbola and the transverse axis AA' as its diameter is called the auxiliary circle of the hyperbola.Another definition of Hyperbola:
A hyperbola can be defined in another way; Locus of a moving point such that the difference of its distances from two fixed points is constant, would be a hyperbola.
Relative Position of a Point with respect to the Hyperbola:
The quantity
is positive, zero or negative, according as the point (x1, y1) lies within, upon or without the curve. It means that for the point (x1, y1) lying outside
the curve, 
> 0 and when lying inside, 
< 0.Parametric Coordinates:
We can express the coordinates of a point of the hyperbola
in terms of a single parameter, say .Any point on the curve, in parametric form is x = a sec
, y = b tan.Another form of parametric equations of the hyperbola is:
x =
.General equation of the second degree:
Consider the general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
It represents a pair of lines if abc + 2fgh - af2 - bg2 - ch2 = 0 i.e.,
= 0 (for two parallel lines,
= 0, h2 = ab),an ellipse if h2 < ab,
0 (for a circle a = b and h = 0),a parabola if h2 = ab,
0a hyperbola if h2 > ab.
0 (for a rectangular hyperbola a + b = 0),
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