Conic Sections | Ellipse & Hyperbola | Chapter-11 | Class XI

Conic Section, Ellipse and its complete explanation with diagrams, Hyperbola and its complete explanation with diagrams

ELLIPSE
An ellipse is the set of all points in a plane sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called foci.

Eccentricity < 1

CENTRE OF THE ELLIPSE:-
 The mid point of the line segment joining the foci is called the centre of the ellipse.

MAJOR AXIS:-
A line segment passing through the foci of the ellipse is called the major axis.

MINOR AXIS:-
A line passing through the centre of the ellipse and is perpendicular to the major axis is called minor axis.

ECCENTRICITY:-
It is defined as the ratio of the half of the  distance between the foci to the length of the semi major axis. Mathematically:-  e = c / a

LATUS RECTUM:-

It is the line segment perpendicular to the major axis through any of  the foci and whose end points lie on the ellipse. Mathematically Length of latus rectum is =\frac{2b^{2}}{a}\]

STANDARD EQUATIONS OF THE ELLIPSE:-

\[When\; major\; axis\; is\; along \; the\; x-axis\; then\; equation \; of \; the\; ellipse\; is\; \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\]

\[When\; major\; axis\; is\; along \; the\; y-axis\; then\; equation \; of \; the\; ellipse\; is\; \frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\]\[c^{2}=a^{2}-b^{2}\]

PROPERTIES OF ELLIPSE

HYPERBOLA:-

It is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is fixed or constant.  Eccentricity > 1

TRANSVERSE AXIS:- 
The line through the focuses is called transverse axis. Length of transverse axis is  2a.

CONJUGATE AXIS:- 
A line through the centre and perpendicular to the transverse axis is called conjugate axis. Length of conjugate axis is  2b

\[Distance\; between\; two\; focuses \; is = 2c ,where\; \; c^{2}=a^{2}+b^{2}\]

ECCENTRICITY:- 
The ratio e = c/a is called eccentricity

LENGTH OF LATUS RECTUM:- 
It is the line segment perpendicular to the transverse axis through any of the focuses and whose end points lie on the hyperbola.

STANDARD EQUATIONS OF HYPERBOLA:-

\[When\; major\; axis \: \: is\: x-axis\: then \: equation\: of\: hyperbola\: is \: \: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\]\[When\; major\; axis \: \: is\: y-axis\: then \: equation\: of\: hyperbola\: is \: \: \frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\]

PROPERTIES OF HYPERBOLA

Art Integrated Project on Conic Section Class 11

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Maths Conic Section Part 2 Class 11 NCERT-cbse mathematics