Conic Sections /Circle & Parabola/Chapter-11/Class-XI

CONIC SECTION

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.

Maths Conic Section Part 1 Class 11 NCERT-cbse mathematics

DIFFERENT TYPES OF CONIC SECTIONS:-
CIRCLE:-  When β(beta) = 90, then intersection of cone with plane is called circle.

ELLIPSE:- When α<β<90 then intersection of plane and cone gives an ellipse.

PARABOLA:- When α= β then intersection of plane and cone gives parabola.

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

Maths Conic Section Part 1 Class 11 NCERT-cbse mathematics

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:-
Let us suppose that a circle of radius O(h,k) and let P(x,y) is any arbitrary point on the circle so that |OP| is the radius of the circle\[|OP|=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\]\[r=\sqrt{(x-h)^{2}+(y-k)^{2}}\]\[r^{2}=(x-h)^{2}+(y-k)^{2}\] This is called the standard form of equation of circle.

Equation of the circle having centre (h, k) and radius = r is \[(x-h)^{2}+(y-k)^{2}=r^{2}\]

When centre is at the origin then equation of the circle is  \[(x-0)^{2}+(y-0)^{2}=r^{2}\; \; or\; \; (x)^{2}+(y)^{2}=r^{2}\]
General Equation of a Circle

\[Equation\; of \; the\; form \; \; x^{2}+y^{2}+2gx+2fy+c=0\; is\; called\; general\; equation\; of\; circle.\]\[Here \; centre \; is\; (-g, -f)\; and \; radius\; is = \sqrt{g^{2}+f^{2}-c}\]\[If \; \; g^{2}+f^{2}-c> 0, then \; the\; radius\; of \; the\; circle\; is\; real\; and\; the \; circle\; is\; also\; real\]\[If \; \; g^{2}+f^{2}-c= 0, then \; the\; radius\; of \; the\; circle\; is\; also\; real\; and\; the \; circle\; is\; a \; point \; circle.\]\[If \; \; g^{2}+f^{2}-c< 0, then \; the\; radius\; of \; the\; circle\; is\;imaginary\; and\; the \; circle\; is\; also\]\[ an\; imaginary\; circle\; and\; it\; is\; not\; possible\; to\; draw\; such \; circle.\]

Special Features of General Equations of a Circle

It is quadratic in both x and y.\[Coefficient\; of\; x^{2}= coefficient\; of\; y^{2}\]

There is no term containing xy.

It contains three arbitrary constants.

PARABOLA 
Maths Conic Section Part 1 Class 11 NCERT-cbse mathematics

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