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PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word “F A T H E R” Given word is : "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

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

Maths Conic Section Part 2 Class 11 NCERT-cbse mathematics 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.

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.

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

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.

Maths Conic Section Part 2 Class 11 NCERT-cbse mathematics

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

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