Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Coordinate Geometry ↓


Conic Sections


Conic sections are the curves obtained by cutting a double cone from a plane. These curves include circles, ellipses, parabolas, and hyperbolas. In coordinate geometry, conic sections are studied by examining their equations and graphs. Let us look at each type of conic section in detail.

Circle

A circle is a set of points on a plane that are equidistant from a fixed point called the center. The constant distance from the center to any point on the circle is called the radius.

The equation of a circle with centre ( (h, k) ) and radius ( r ) is:

(x - h)^2 + (y - k)^2 = r^2

When the center of the circle is at the origin ( (0, 0) ), the equation simplifies to:

x^2 + y^2 = r^2
R (H, K)

Ellipse

An ellipse is a set of points where the sum of the distances from two fixed points (called foci) is constant. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest.

The standard form of an ellipse with centre at ( (h, k) ) with principal axis along the x-axis is:

(frac{(x - h)^2}{a^2} + frac{(y - k)^2}{b^2} = 1)

Where ( a ) is the semi-major axis and ( b ) is the semi-minor axis. If ( a > b ), then the major axis is horizontal. If ( b > a ), then it is vertical.

A B

Parabola

A parabola is a set of points such that any point on the parabola is equidistant from a fixed point called the focus and a fixed line called the directrix.

The standard form of a parabola with vertex at ( (h, k) ) can vary depending on its orientation:

  • Vertical axis: ( y = a(x - h)^2 + k )
  • Horizontal axis: ( x = a(y - k)^2 + h )
Center

Hyperbola

A hyperbola is a set of points where the difference in distance from two fixed points (called foci) is constant. It has two branches, each pointing away from the center.

The standard form of a hyperbola with center at ( (h, k) ) and transverse axis along the x-axis is:

(frac{(x - h)^2}{a^2} - frac{(y - k)^2}{b^2} = 1)

If the transverse axis is along the y-axis, then the equation is:

(frac{(y - k)^2}{a^2} - frac{(x - h)^2}{b^2} = 1)

Properties of conic section

Each type of conic section exhibits the following distinctive features:

  • Circle: no focus, constant radius.
  • Ellipse: two focuses, centered at the midpoint of the major axis, affine distance sum constant.
  • Parabola: Single focus and directrix, equally spaced vertices.
  • Hyperbola: Two divergent branches, two focuses, constant difference of distances.

Applications of conic section

Conic sections have diverse applications in fields such as engineering, physics, astronomy, and architecture. For example:

  • Elliptical: The orbits of the planets are elliptical, governed by Kepler's laws.
  • Parabola: Parabolic mirrors and antennas focus waves to a point.
  • Radio navigation and determination of locations are carried out using the hyperbolic theory.

Summary

Conic sections are an essential part of coordinate geometry. Each cone has unique properties and can serve practical purposes in real-life situations. Understanding the equations and characteristics of these curves increases knowledge about their geometric nature and applications.


Grade 11 → 7.2


U
username
0%
completed in Grade 11

← Prev (7.1.5)
Equation of Lines
Next (7.2.1) →
Circle

Comments 



Conic Sections

https://www.buddymath.com/g11/7.2?bmal=en