NCERT Class 11-Math's: Exemplar Chapter –11 Conic Sections Part 2

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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.

Image of Parabola

Image of Parabola

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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

Standard Equations of Parabola Fig 11.7

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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

Main Facts About the Ellipse Fig. 11.8

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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

Conics

Parametric equation

(i)

Parabola

(ii)

Ellipse:

(iii)

Hyperbola: