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 ↓


Intercept Form


In coordinate geometry, there are many ways to represent a line graphically. One such simple and effective way is through the intercept form of a line. The intercept form of a line gives you a straightforward means of visualizing and understanding linear equations. It shows the points where the line crosses the axes, making graph reading and line plotting much easier to understand. Let us dive into the comprehensive understanding of the intercept form of a line.

What is intercept form?

In coordinate geometry, the line intercept form is a way of describing the equation of a line in terms of its x-intercept and y-intercept. The standard form of a line equation in this form is:

x/a + y/b = 1

Here, a represents the x-intercept, which is the point where the line crosses the x-axis. Similarly, b represents the y-intercept, which is the point where the line crosses the y-axis.

This form is particularly useful because it shows the constraints directly and allows for easy graphing and visualization. To look at this in more detail, let's break down each part of the equation:

  1. x-intercept (a): It is the value of x when y = 0 At this point, the line crosses the x-axis. Therefore, the coordinates of the x-intercept are (a, 0).
  2. y-intercept (b): It is the value of y when x = 0 At this point, the line crosses the y-axis. Therefore, the coordinates of the y-intercept are (0, b).

Obtaining the intercept form

To understand how the intercept form is derived, let's start with the general form of a linear equation:

Ax + By + C = 0

Rearranging this equation, moving 'C' to the other side, we get:

Ax + By = -C

We want to express this in intercept form, x/a + y/b = 1 To do this, we first need to make the right side of the equation equal to 1. To do this, divide the entire equation by -C:

(Ax)/(-C) + (By)/(-C) = 1

In this form, substitute -C/A for a and -C/B b, and we get:

x/a + y/b = 1

This equation is now in intercept form, where:

  • a = -C/A
  • b = -C/B

Visual understanding using graphs

Let's understand the intercept form visually with some interactive examples:

X Y (a,0) (0,b)

This line intersects the x-axis at (a, 0) and the y-axis at (0, b).

This graphical representation makes it more intuitive to understand the intercept form. As discussed, an important aspect of this representation is that it highlights the specific points where the line crosses the axes.

Example to understand intercept form

Some text examples with solutions are given below to enhance your understanding about the intercept form of a line:

Example 1: Convert the equation 3x + 4y = 12 into intercept form.

Solution: Start by rearranging it:

  • 3x + 4y = 12 → 3x/12 + 4y/12 = 1
  • Simplify: (x/4) + (y/3) = 1

So, the intercept form is x/4 + y/3 = 1 Here, the x-intercept (a) is 4, and the y-intercept (b) is 3.

Example 2: If the x-intercept of a line is 5 and y-intercept is -7, then write the equation in intercept form.

Solution: Plug the intercept values directly:

  • x/5 + y/(-7) = 1

The intercept of the line is of the form x/5 - y/7 = 1.

Application of the intercept form

Understanding the intercept pattern can be applied in a variety of practical scenarios and mathematical problem-solving situations:

  • Understanding linear relationships: You can easily find out the linear relationship between two quantities.
  • Graphical plots: Quick plotting of graphs with intercepts, helps evaluate intersections and slopes.
  • Problem solving: Helps in understanding and correcting constraints in equations of physics, chemistry and even economics.

Benefits of intercept form

There are several distinct advantages to using the intercept form:

  • Simplicity and intuition: Provides the x-intercept and y-intercept directly, making it intuitive and useful for graphical analysis.
  • Graphing: Useful for quickly graphing a line once the intercepts are known.
  • Ease of calculation: Easy handling of the equation, which can also aid in calculations related to parallel and perpendicular lines.

Challenges in using the intercept form

Although the blockchain format is simple and intuitive, it still has some challenges:

  • Undefined intercepts: If a line is parallel to one of the axes, then the intercept form for the missing intercept becomes undefined.
  • Non-vertical/horizontal lines: In cases other than vertical and horizontal lines, additional steps are sometimes necessary to convert other forms to intercept forms.

Conclusion

The intercept form is an important concept for learners and practitioners of coordinate geometry because it brings directness and clarity to the graphical representation of lines. By emphasizing the understanding of how lines interact with the coordinate axes, this form simplifies the framework necessary for further exploration in geometry and algebra. In short, mastering the intercept form of a line provides a foundation for more complex problem-solving and aids in the intuitive understanding of geometry.


Grade 10 → 3.6.4


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Two-Point Form
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General Form of a Line

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Intercept Form

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