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 ↓


Properties of Conics


In mathematics, particularly in coordinate geometry, conic sections or cones are shapes that can be formed as the intersection of a plane and a cone. Their various properties make conic sections a fascinating topic that finds many applications in the realm of pure mathematics and real-world applications. These segments can be classified as ellipses, parabolas, and hyperbolas, and each of them has different characteristics.

Definitions of conic section

Before considering the properties of each cone, it is important to understand what these shapes are:

  • Ellipse: An ellipse looks like a flattened circle. It is the set of all points where the sum of the distances from two fixed points, called foci, is constant.
  • Parabola: A parabola is a mirror-symmetric curve where any point is equidistant from the focus (a fixed point) and the directrix (a fixed straight line).
  • Hyperbola: A hyperbola consists of two mirrored curves, called branches, that diverge. It is the set of all points where the absolute difference of the distances to the two foci is constant.

General equation of cones

Conic sections can be understood through their general quadratic equation as:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The shape of the cone depends on the values of the coefficients A, B, and C. Here's how the relationship between these coefficients determines the type of cone:

  • Ellipse and Circle: When B² - 4AC < 0
  • Parabola: When B² - 4AC = 0
  • Hyperbola: When B² - 4AC > 0

Properties of ellipse

Some interesting properties of the ellipse are:

The standard equation of an ellipse centered at the origin is given by:

(x²/a²) + (y²/b²) = 1

where a is the semi-major axis and b is the semi-minor axis. The length of the semi-major axis is always equal to or greater than the semi-minor axis (a ≥ b).

Visual example of an ellipse

B A

The foci of an ellipse are two fixed points on the principal axis. The sum of the distances from any point on the ellipse to the two foci is constant. The distance from the center to each focus, denoted as c, is calculated as:

c² = a² - b²

Properties of parabolas

The parabola has several unique properties:

The standard equation of a parabola with vertex at the origin and axis of symmetry along the y-axis is:

y² = 4ax

where a is the distance from the vertex to the focus.

Visual example of a parabola

Center

In this form, the directrix is a line parallel to the y-axis, located at a distance a units from the vertex on the side opposite to the focus.

Properties of hyperbola

Hyperbolas are characterized by their symmetrical open curves:

The standard equation of a hyperbola centered at the origin is:

(x²/a²) - (y²/b²) = 1

where a is the distance from the center to the vertices along the x-axis, and b relates to the distance to the asymptotes.

Visual example of hyperbola

A hyperbola has two branches, and the asymptotes are the lines that the branches approach but never touch. The distance of each focus from the center is c, and is found by:

c² = a² + b²

Text examples and applications

Let's look at how these conic sections might appear in real-world scenarios:

Example 1: Ellipse in astronomy

According to Kepler's first law of planetary motion, the orbits of planets around the Sun are ellipses, with the Sun at one focus. Suppose the semi-major axis ((a)) for a planet's orbit is 100 million km, and the semi-minor axis ((b)) is 80 million km.

Substituting into the equation c² = a² - b², we find the focal length:

c² = (100)² - (80)² = 10000 - 6400 = 3600
c = √3600 = 60

Therefore, each focus is 60 million km away from the centre.

Example 2: Parabola in design

Parabolic shapes are commonly used in satellite dishes. Suppose you are designing a dish such that the focus is 5 units away from the vertex. The standard form would be:

y² = 4 * 5 * x = 20x

Summary

Conic sections – ellipses, parabolas and hyperbolas – are complex mathematical shapes with specific properties. Each of them plays a vital role in understanding the mathematics behind various natural and manufactured phenomena. From the orbits of celestial bodies to architectural designs, these shapes exist in abundance, finding a strong place in theoretical mathematics as well as real-world applications.


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Properties of Conics

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