Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Coordinate Geometry → Equation of a Line ↓


Two-Point Form


In coordinate geometry, one of the fundamental aspects is to understand how to derive the equation of a line. There are various forms of line equations, but among them, the “two-point form” is highly useful, especially when you know two points through which a line passes. This method allows you to derive the equation using only these two points.

Understanding the concept

The two-point form of the equation of a line helps you express the line algebraically. If you have two points on a line, say (x 1, y 1) and (x 2, y 2), you can use them to find the equation of the line that connects these two points.

Mathematical formulas

The formula for the two-point form is derived from the concept of slope, which measures the steepness or inclination of a line. The slope m of a line passing through two points (x 1, y 1) and (x 2, y 2) is calculated as follows:

M = (Y 2 - Y 1) / (X 2 - X 1)

Using this slope, the two-point form of the equation of the line is:

y - y 1 = m(x - x 1)

Substituting m in the equation, we get:

y - y 1 = ((y 2 - y 1) / (x 2 - x 1))(x - x 1)

This is the two-point form of the equation of a line.

Example 1

Consider two points on a line: (2, 3) and (4, 7) We will use these points to find the equation of the line.

Step 1: Calculate the slope (m)

m = (7 – 3) / (4 – 2) = 4 / 2 = 2

Step 2: Use the two-point form of the line equation with the point (2, 3):

y – 3 = 2(x – 2)

Step 3: Simplify the equation:

y – 3 = 2x – 4

Add 3 to both sides to solve for y:

y = 2x – 1

Therefore, the equation of the line passing through the points (2, 3) and (4, 7) is y = 2x - 1.

(2, 3) (4, 7)

The line diagram given above shows the line passing through the points (2, 3) and (4, 7) marked on the Cartesian plane. The points are marked, and you can see the line that represents the equation y = 2x - 1.

Example 2

Let's look at another example. Find the equation of the line passing through (-1, 5) and (3, -1).

Step 1: Calculate the slope (m)

m = (-1 - 5) / (3 - (-1)) = -6 / 4 = -3/2

Step 2: Use the two-point form of the line equation with the point (-1, 5):

y – 5 = (-3/2)(x + 1)

Step 3: Simplify the equation:

y – 5 = (-3/2)x – 3/2

Add 5 to both sides to solve for y:

y = (-3/2)x + 7/2

Hence, the equation of the line passing through the points (-1, 5) and (3, -1) is y = (-3/2)x + 7/2.

(-1, 5) (3, -1)

The line above shows a negative slope. The points (-1, 5) and (3, -1) are plotted, which shows the line described by y = (-3/2)x + 7/2.

Why two-point form is useful

Two-point form is incredibly useful for many reasons. It provides a simple yet powerful way to understand and analyze lines. Whether you're working on graphing applications, solving systems of equations, or understanding geometric shapes, it provides a solid foundation for calculations. It helps you transform two known points directly into an equation that defines a consistent relationship throughout the graph.

Some of its useful applications are as follows:

  • Forecasting trends using known data points in real-world scenarios, such as economics or physics.
  • Modeling and understanding geometric shapes or designing graphs and charts.
  • Simplifying complex functions and equations in mathematical problem-solving.

Important things to remember

  1. The points must be clear and distinct, otherwise the slope will be undefined.
  2. Make sure both points are accurate so that the required correct line can be drawn.
  3. This form can be converted into other forms, such as the slope-intercept form, which is especially useful for graphing and intercept calculations.

Conclusion

Two-point form is a simple method of finding the equation of a line when two points are known. Mastering this form significantly aids in understanding coordinate geometry, leading to stronger problem-solving abilities when dealing with linear relationships.


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