Conic Sections | Circle & Parabola | Class-11 | Ch-10

Conic Sections, Circle & Parabola,
Chapter-10, Class-XI

The sections of the double napped right circular cone with the plane are called conic sections. eg:- circle, ellipse, parabola and hyperbola etc. all are the examples of conic sections.

APPLICATIONS:-
These curves have wide applications in the field of planetary motion, design of telescope, antenas, reflectors of flash light and automobile headlights.

Vertex of the double napped cone separate the cone into two parts each part is called Nappes.
α (alpha) is the angle made by the generator with the x-axis.

DIFFERENT TYPES OF CONIC SECTIONS:-

CIRCLE:-  

When β(beta) = 90, then intersection of cone with plane is called circle. For circle the value of eccentricity = 0 or  e = 0

PARABOLA:- 

When α= β then intersection of plane and cone gives parabola. For Parabola the value of eccentricity = 1 or  e = 1
ELLIPSE:- 

When α < β < 90 then intersection of plane and cone gives an ellipse. For ellipse the value of eccentricity < 1 or  e < 1
HYPERBOLA:- 

When 0 < β < α then the intersection of plane and cone gives a hyperbola.

For Hyperbola the value of eccentricity > 1 or  e > 1

CIRCLE:- 

A circle is the set of all points in the plane which are at equidistant from the fixed point in the plane. Fixed point is called the centre of the circle, and fixed distance is called the radius of the circle.

EQUATION OF THE CIRCLE in STANDARD FORM

(a) Central form of equation of circle

Let us suppose that a circle of centre O(h, k) and let P(x, y) is any arbitrary point on the circle so that |OP| is the radius of the circle

       

     

 This is called the standard form of equation of circle.

Equation of the circle having centre (h, k) and radius = r is 

(b) Simplest form of equation of circle

When centre is at the origin then equation of the circle is  

(x - 0)² + (y - 0)² = r²  or x² + y² = r²

(c) Diameter form of equation of circle.

AB is the diameter of the circle with centre c. P(x, y) is any point on the circle then equation of the circle is 

(x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0

General Equation of a Circle

Equation of the form x² + y² + 2gx + 2fy + c = 0 is called the general equation of the circle. 

Centre of the ciecle is (-g, -f) or 

Radius of the circle is given by

If g² + f² – c > 0, then the radius of the circle is real and the circle is also real.

If g² + f² – c = 0, then the radius of the circle is real and the circle is called a point circle and is called degenerate circle.

If g² + f² – c < 0, then the radius of the circle is imaginary and the circle is called is also an imaginary circle and it is not possible to draw an imaginary circle. 

 

Special Features of General Equations of a Circle

It is quadratic in both x and y.

Coefficient of x² = Coefficient of y² 

There is no term containing xy.

It contains three arbitrary constants g, f and c

PARABOLA 

A parabola is the set of all points in a plane which are at equidistant from the fixed line and the fixed point.
Para means  "for"  and  bola means  "throwing"

The fixed line is called the directrix and the fixed point is called the focus.

AXIS:-
 A line through the focus and perpendicular to the directrix is called the axis of the parabola.

VERTEX :- 
The point of intersection of the parabola with the axis is called the vertex of the parabola.

LATUS RECTUM:-
Latus 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.

Art Integrated Project on Conic Section Class 11

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Maths Conic Section Part 1 Class 11 NCERT-cbse mathematics