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 → Straight Lines ↓


Equation of Lines


In coordinate geometry, the equation of a line is a fundamental concept used to describe a straight line in a plane. Understanding the equation of lines is important because it helps us represent and analyze geometric shapes and solve problems related to them. In this article, we will learn about the equations of lines in detail, including the different forms of line equations, how to derive them, and how to use them.

What is Rekha?

A line is a straight one-dimensional figure that has no thickness and extends infinitely in both directions. It is determined by at least two points. In the coordinate plane, we usually specify a line by identifying its slope and y-intercept or by knowing two distinct points on it.

Slope intercept form

The slope-intercept form is one of the simplest and most widely used forms of a line equation. It is expressed as:

y = mx + b

Where:

  • m is the slope of the line.
  • b is the y-intercept of the line, which is the point where the line intersects the y-axis.

Slope (m) represents the change in the y-coordinate for a one-unit change in the x-coordinate. It is calculated as the ratio of the rise (change in y) over the run (change in x). If the slope is positive, the line goes up from left to right; if it is negative, the line goes down.

Visual example:

y = mx + b 0 X Y

Point-slope form

The point-slope form is useful when you know a point on the line and the slope. It is presented like this:

y - y1 = m(x - x1)

Where:

  • (x1, y1) is a known point on the line.
  • m is the slope of the line.

This form is especially useful for quickly finding the equation of a line when the points on the line and the slope are known.

Text example:

Suppose we have a point (3, 2) and slope 4. The point-slope form of the line can be written as:

y - 2 = 4(x - 3)

Expanding this equation gives the result:

y = 4x - 12 + 2

Thus, the equation becomes:

y = 4x - 10

General form of the line

The normal form of a line is a straightforward way to express the line equation in a standard format:

Ax + By + C = 0

Here:

  • A, B and C are constants.
  • Both A and B are not zero.

Although this form is less intuitive than the slope-intercept form, it can still be useful because it represents all lines, including vertical lines, which cannot be expressed as y = mx + b.

Text example:

If we have the line equation 3x + 4y - 12 = 0, we can rearrange it into slope-intercept form to find its slope and y-intercept:

4y = -3x + 12
y = -3/4x + 3

Vertical and horizontal lines

Vertical and horizontal lines are special cases of straight lines with unique equations.

Vertical lines

A vertical line is parallel to the y-axis and its slope is undefined. The equation of the vertical line is:

x = a

where a is the x-coordinate of all points on the line.

Horizontal lines

A horizontal line is parallel to the x-axis and has a slope of zero. The equation of the horizontal line is:

y = b

where b is the y-coordinate of all points on the line.

Visual example:

y = b x = a 0 X Y

Two-point form

This form is needed when you don't know the slope but have two points (x1, y1) and (x2, y2) on the line. It is expressed as follows:

y - y1 = ((y2 - y1)/(x2 - x1)) * (x - x1)

This equation basically calculates the slope by dividing the difference in the y-values by the difference in the x-values.

Text example:

For two points (2, 3) and (5, 11), plug the values in the formula and get:

y - 3 = ((11 - 3)/(5 - 2)) * (x - 2)

Since ((11 - 3)/(5 - 2)) = 8/3, the equation becomes:

y - 3 = (8/3)(x - 2)

By expanding it we get:

y = (8/3)x - 16/3 + 3
y = (8/3)x + 9/3
y = (8/3)x + 3

Correlations between different forms

Understanding the different forms of line equations helps you modify them to suit your needs. For example, converting point-slope form to slope-intercept form can make it easier to identify the slope and y-intercept.

Text conversion example:

Suppose we start with the point-slope form of a line: y - 2 = 5(x - 1). To convert this to slope-intercept form:

y - 2 = 5x - 5
y = 5x - 5 + 2
y = 5x - 3

From the transformation it is clear that the slope m is 5, and the intercept b is -3. Such transformations simplify the visualization and understanding of the behavior of the line on the graph.

Conclusion

The equation of a line in coordinate geometry encompasses many different forms, each of which has its own utility that requires different data points. Mastering these forms enriches your geometric understanding and equips you with versatile tools for solving complex problems involving lines. Whether working with the simplicity of the slope-intercept form, the uniqueness of the point-slope or two-point forms, or the generality of the standard form, these equations are foundational to coordinate geometry.


Grade 11 → 7.1.5


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Equation of Lines

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