Topics
Angle and Its Measurement
- Directed Angle
- Angles of Different Measurements
- Angles in Standard Position
- Measures of Angles with Various Systems
- Area of a Sector
- Length of an Arc
Trigonometry - 1
- Trigonometric Ratios
- Trigonometric Functions with the Help of a Circle
- Signs of Trigonometric Functions in Different Quadrants
- Range of Cosθ and Sinθ
- Trigonometric Functions of Specific Angles
- Trigonometric Functions of Negative Angles
- Important Identities and Standard Results
- Periodicity of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Graphs of Trigonometric Functions
- Polar Co-ordinate System
Trigonometry - 2
Determinants and Matrices
- Definition and Expansion of Determinants
- Minors and Cofactors of Elements of Determinants
- Properties of Determinants
- Application of Determinants
- Determinant Method (Cramer’s Rule)
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Concept of Matrices
- Types of Matrices
- Operation on Matrices
- Properties of Matrix Multiplication
- Transpose of a Matrix
Straight Line
- Locus of a Points in a Co-ordinate Plane
- Equations of Line in Different Forms
- Family & Concurrent Lines
Circle
Conic Sections
Measures of Dispersion
- Meaning and Definition of Dispersion
- Measures of Dispersion
- Quartiles and Range in Statistics
- Variance
- Standard Deviation
- Change of Origin and Scale of Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Probability
Complex Numbers
Sequences and Series
- Sequence, Series, and Progression
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Harmonic Progression (H. P.)
- Arithmetico Geometric Series
- Power Series
Permutations and Combination
Methods of Induction and Binomial Theorem
- Principle of Mathematical Induction
- Binomial Theorem for Positive Integral Index
- General Term in Expansion of (a + b)n
- Middle term(s) in the expansion of (a + b)n
- Binomial Theorem for Negative Index Or Fraction
- Binomial Coefficients
Sets and Relations
- Sets and Their Representations
- Classification of Sets
- Fundamental Concepts of Ordered Pairs and Relations
- Intervals
Functions
Limits
Continuity
Differentiation
- Definition of Derivative and Differentiability
- Rules of Differentiation (Without Proof)
- Derivative of Algebraic Functions
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivatives of Exponential Functions
- L' Hospital'S Theorem
Estimated time: 5 minutes
Maharashtra State Board: Class 12
Definition: Hyperbola
A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point (i.e. focus) to its distance from a fixed line (i.e. directrix) is always constant and greater than unity.
Maharashtra State Board: Class 12
Key Points: Hyperbola
| Property |
Standard Hyperbola \[\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\] |
Conjugate Hyperbola \[\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1\] |
|---|---|---|
| Centre | (0, 0) | (0, 0) |
| Vertices | (±a, 0) | (0, ±b) |
| Transverse Axis Length | 2a | 2b |
| Conjugate Axis Length | 2b | 2a |
| Foci | (±ae, 0) | (0, ±be) |
| Eccentricity | \[\mathrm{e}=\frac{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}{\mathrm{a}}\] | \[\mathrm{e}=\frac{\sqrt{\mathrm{b}^2+\mathrm{a}^2}}{\mathrm{b}}\] |
| Relation | \[\mathbf{b}^2=\mathbf{a}^2(\mathbf{e}^2-1)\] | \[\mathbf{a}^2=\mathbf{b}^2(\mathbf{e}^2-1)\] |
| Directrices | \[x=\pm\frac{\mathrm{a}}{\mathrm{e}}\] | \[y=\pm\frac{\mathrm{b}}{\mathrm{e}}\] |
| Length of Latus Rectum | \[\frac{2\mathrm{b}^2}{\mathrm{a}}\] | \[\frac{2\mathrm{a}^2}{\mathrm{b}}\] |
| Ends of Latus Rectum | \[\left(\pm ae,\pm\frac{b^{2}}{a}\right)\] | \[\left(\pm\frac{a^{2}}{b},\pm e\right)\] |
| Distance between Foci | 2ae | 2be |
| Difference of Focal Radii | 2a | 2b |
| Axis Equations | Transverse: y = 0, Conjugate: x = 0 | Transverse: x = 0, Conjugate: y = 0 |
| Parametric Equations | x = a secθ, y = b tanθ | x = a tanθ, y = b secθ |
| Parametric Point | (a secθ, b tanθ) | (a tanθ, b secθ) |
| Tangent at Vertex | x = ±a | y = ±b |
Shaalaa.com | Standard Equation of Hyperbola
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