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
- Expansion of Determinant
- 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
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> 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
- Basics of Relations & Functions
- Intervals
Functions
Limits
Continuity
Differentiation
- Concept of Differentiability
- Rules of Differentiation (Without Proof)
- Derivative of Algebraic Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Exponential and Logarithmic Functions
- L' Hospital'S Theorem
Estimated time: 4 minutes
Maharashtra State Board: Class 12
Key Points: Equation of Tangent and Condition of Tangency
For Standard Circle: x² + y² = a²
| Sr. No. | Description | Formula |
|---|---|---|
| i. | Tangent at a point (x₁, y₁) | xx₁ + yy₁ = a² |
| ii. | Parametric form of tangent at P(θ) | x cosθ + y sinθ = a |
| iii. | Condition of tangency for the line y = mx + c | \[\mathrm{c=\pm a~\sqrt{1+m^{2}}}\] |
| Point of contact | \[\left(\frac{-\mathrm{a}^{2}\mathrm{m}}{\mathrm{c}},\frac{\mathrm{a}^{2}}{\mathrm{c}}\right)\] | |
| iv. | Equation of tangent in terms of its slope m | \[y=\mathrm{m}x\pm\mathrm{a}\sqrt{1+\mathrm{m}^{2}}\] |
| v. | Length of tangent from the point (x₁, y₁) | \[\sqrt{S_{1}}=\sqrt{x_{1}^{2}+y_{1}^{2}-a^{2}}\] |
| vi. | Equation of the Director circle | x² + y² = 2a² |
For General Circle: x² + y² + 2gx + 2fy + c = 0
| Sr. No. | Description | Formula |
|---|---|---|
| i. | Tangent at a point (x₁, y₁) | xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 |
| ii. | Length of tangent from the point (x₁, y₁) | \[\sqrt{S_{1}}=\sqrt{x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c}\] |
Number of Common Tangents:
| Case | Diagram | No. of Tangents | Condition |
|---|---|---|---|
| Disjoint circles |  |
4 | d > r₁ + r₂ |
| Touch externally |  |
3 | d = r₁ + r₂ |
| Intersecting circles |  |
2 | d < r₁ + r₂ |
| Touch internally |  |
1 | d = \[\left|\mathbf{R}_{1}-\mathbf{R}_{2}\right|\] |
| Concentric circles |  |
0 | d = 0 |
Equation of a pair of tangents:
(x² + y² − a²)(x₁² + y₁² − a²) = (xx₁ + yy₁ − a²)²
