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 ↓


Circle


In mathematics, especially coordinate geometry, the circle is a fascinating figure that is often studied in Class 11. A circle is a simple closed figure that is the set of all points in a plane that are a fixed distance from a fixed point. This fixed point is called the center of the circle, and the distance from the center to any point on the circle is called the radius.

Definition of circle

A circle can be defined as the locus of points in the plane equidistant from a given point. Mathematically, if the center of a circle is at the point ( (h, k) ) and the radius is ( r ), then a point ( (x, y) ) lies on the circle if and only if:

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

This equation is called the standard equation of a circle. Let's break down each component of the equation:

  • Centre: The point ( (h, k) ) is the centre of the circle.
  • Radius: The value ( r ) represents the radius of the circle, which is the distance from the center to any point on the circle.
  • Point ( (x, y): Any point that satisfies the equation lies on the circumference of the circle.

Equation of a circle

Standard form

If the center of a circle is at the origin ( (0, 0) ) and the radius is ( r ), then the equation simplifies to:

x^2 + y^2 = r^2
(0,0) radius r

In this particular case, the center of the circle is the origin, so the equation has no ( h ) and ( k ) components.

General form

The general form of the equation of a circle is given as:

x^2 + y^2 + 2gx + 2fy + c = 0

In this equation:

  • ( g ) and ( f ) are constants related to the center of the circle.
  • ( c ) is a constant that affects the distance of the circle from the origin.
The center ( (h, k) ) and radius ( r ) of a circle can be determined from ( g ), ( f ), and ( c ) by the following relations:

h = -g, k = -f, r = sqrt{g^2 + f^2 - c}

Example

Example 1: Circle with known center and radius

Let us consider a circle with center ( (3, -2) ) and radius 5. To find the equation of the circle, substitute these values into the standard form equation ( (x - h)^2 + (y - k)^2 = r^2 ):

(x - 3)^2 + (y + 2)^2 = 25

Expanding this equation gives:

x^2 - 6x + 9 + y^2 + 4y + 4 = 25

x^2 + y^2 - 6x + 4y - 12 = 0

Thus, the equation of the circle is ( x^2 + y^2 - 6x + 4y - 12 = 0 ).

Example 2: Circle with center at the origin

Consider a circle with center at the origin and radius 7. The equation of this circle is:

x^2 + y^2 = 49
(0,0) 7

The simplicity of this equation arises from the fact that there is no horizontal or vertical shift from the origin.

Properties of circle

Understanding the basic properties of a circle is essential to solve problems related to this figure in coordinate geometry. Here are some key properties:

  • Symmetry: A circle is symmetric about its center. It looks the same from every direction, making its study very interesting in geometry.
  • Chord: A line segment joining two points on a circle. The longest chord is the diameter.
  • Diameter: A special type of chord that passes through the center. It is twice the radius: ( d = 2r ).
  • Circumference: The total distance around a circle, calculated using ( C = 2pi r ).
  • Area: The area enclosed by a circle, given by the formula ( A = pi r^2 ).

Relation with conic sections

Circles as conic sections can be obtained by intersecting a right circular cone with a plane parallel to its base. The intersection curve is a circle. This makes the circle a special case of an ellipse where the two focal points meet.

(H,K)

Examples of circles in coordinate geometry

Example 3: Finding the center and radius from the equation

Given a circle equation in general form: ( x^2 + y^2 - 4x + 6y + 9 = 0 ), find the center and radius.

First, rewrite the equation by completing the square:

x^2 - 4x + y^2 + 6y = -9

Complete the square for the ( x ) and ( y ) terms:

(x^2 - 4x + 4) + (y^2 + 6y + 9) = -9 + 4 + 9

(x - 2)^2 + (y + 3)^2 = 4

From this we identify the center as ( (2, -3) ) and the radius as ( sqrt{4} = 2 ).

Example 4: Converting the equation of a circle from general to standard form

Consider the circle given by the equation ( 2x^2 + 2y^2 - 8x + 16y - 4 = 0 ).

Divide by 2 to simplify the entire equation:

x^2 + y^2 - 4x + 8y - 2 = 0

Rewrite the square by completing it:

(x^2 - 4x + 4) + (y^2 + 8y + 16) = 2 + 4 + 16

(x - 2)^2 + (y + 4)^2 = 22

Thus, the circle centered at ( (2, -4) ) has radius ( sqrt{22} ).

Visual path of locus formation

Imagine placing one end of a thread at a fixed point (the center) and drawing a complete circle around it. Each position of the tip of the thread, when it is pulled taut, marks the circumference of the circle, demonstrating the concept of a circle as a locus of points.

hideout Center

Conclusion

Understanding circles in coordinate geometry involves exploring different forms of equations, properties, and their geometric implications. As can be seen from the examples, being able to move between standard and normal forms, completing the square, and identifying key components such as radius and center are essential skills. Mastering these concepts makes it possible to accurately represent and manipulate circles in a variety of mathematical contexts. In practice, these principles form the basis for more complex geometric and algebraic applications, illustrating the beauty and utility of circles within mathematics and beyond.


Grade 11 → 7.2.1


U
username
0%
completed in Grade 11

← Prev (7.2)
Conic Sections
Next (7.2.2) →
Parabola

Comments 



Circle

https://www.buddymath.com/g11/7.2.1?bmal=en