NCERT Class 11-Math՚S: Exemplar Chapter – 11 Conic Sections Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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11.1. 3 Parabola

A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distances from a fixed line l in the plane. The fixed point F is called focus and the fixed line l the directrix of the parabola.

Parabola

Standard equations of parabola

The four possible forms of parabola are shown below in Fig. 11.7 (a) to (d) . The lotus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola (Fig. 11.7) .

Standard Equations of Parabola Fig 11.7

Main facts about the parabola

Main Facts About the Parabola
Forms of Parabolas
Axis
Directrix
Vertex
Focus
Length of lotus rectum
Equations of latus rectum

Focal distance of a point

Let the equation of the parabola be and be a point on it. Then the distance of P from the focus is called the focal distance of the point, i.e.. ,

11.1. 4 Ellipse

An ellipse is the set of points in a plane, the sum of whose distances from two fixed points is constant. Alternatively, an ellipse is the set of all points in the plane whose distances from a fixed point in the plane bears a constant ratio, less than, to their distance from a fixed line in the plane. The fixed point is called focus, the fixed line a directrix and the constant ratio (e) the centricity of the ellipse.

We have two standard forms of the ellipse, i.e.. ,

(i)

In both cases and .

In (i) major axis is along x-axis and minor along y-axis and in (ii) major axis is along and minor along x-axis as shown in Fig. 11.8 (a) and (b) respectively.

Main facts about the Ellipse:

Main Facts About the Ellipse Fig. 11.8
Focal Distre
Forms of the ellipse
Equation of major axis
Length of major axis
Equation of Minor axis
Length of Minor axis
Directories
Equation of latus rectum
Length of latus rectum
Centre

Focal Distance:

The focal distance of a point on the ellipse is

from the nearer focus

from the farther focus

Differences of the focal distances of any point on a hyperbola is constant and equal to the length of the transverse axis.

Parametric equation of conics:

Parametric Equation of Conics
ConicsParametric equation
(i)Parabola
(ii)Ellipse:
(iii)Hyperbola: