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 ↓


Point-Slope Form


In coordinate geometry, one of the major topics you will come across is the equation of a line. There are several forms in which we can express a line equation, and one of these is the point-slope form. The point-slope form is particularly useful when you know a point on a line and the slope of the line. Let’s dive deep into this concept and understand it thoroughly.

Understanding the basics

Before exploring the point-slope form, it's important to understand two basic concepts: point and slope.

What is a point?

In a two-dimensional plane, a point is represented by a set of coordinates (x, y). For example, the point (3, 5) means that you move 3 units along the x-axis and 5 units along the y-axis.

What is a slope?

The slope of a line measures how steep the line is. Mathematically, it is defined as "run over rise," which is the change in the y-coordinate divided by the change in the x-coordinate between two points on a line. If you have two points (x1, y1) and (x2, y2), the slope m is calculated as:

M = (Y2 - Y1) / (X2 - X1)

Point-slope formulation

The point-slope form of a line is represented as:

y - y1 = m(x - x1)

Here, (x1, y1) is a specific point on the line, and m is the slope of the line. This format allows you to write the equation of a line when you know a point on the line and the slope.

Why use the point-slope form?

The point-slope form is especially useful when you are given a point and a slope on a line and need to write the equation of the line. It also provides a straightforward way to see changes in the line based on a change in the slope or point.

Visualization of the point-slope form

Let's look at the point-slope form using a simple graphical example. Consider the following scenario:

(4, 3) Line: y - 3 = 1/2(x - 4) 1 2

In this example, the red dot represents the point (4, 3) on the line. The slope is m 1/2, which indicates that for every 2 units you move horizontally along the x-axis, you move 1 unit vertically along the y-axis. Thus, the equation of the line is:

y – 3 = 1/2(x – 4)

Step-by-step explanation

Let's take a step-by-step approach to understanding how to use the point-slope form.

1. Identify the point

Find the point the line passes through. This point will have coordinates (x1, y1). For example, let's say you have a point (2, 3).

2. Determine the slope

Identify the slope of the line. This value can be provided or calculated if you have two points. Let's assume the slope m is 4.

3. Plug in the point-slope formula

Enter the values in point-slope form:

y - y1 = m(x - x1)

Substituting the values we have, we get:

y – 3 = 4(x – 2)

4. Simplify the equation

You can simplify this equation into slope-intercept form y = mx + b if necessary:

y – 3 = 4(x – 2)
y – 3 = 4x – 8
y = 4x – 8 + 3
y = 4x – 5

Now, you have the slope-intercept form of the line, which is handy for graphing and understanding the direction of the line.

Changes and variations

Sometimes, it is important to understand how changes in slope and point affect the line. Let's look at some changes through examples.

Changes in slope

If the slope increases, the line becomes steeper. Consider if the slope changes from 1/2 to 2 while keeping the point constant:

Root: y - 3 = 1/2(x - 4)
New: y - 3 = 2(x - 4)

Change in point

Changing the point moves the line in the coordinate plane. Suppose you change the point from (4, 3) to (1, 1) while keeping the slope constant:

Root: y - 3 = 1/2(x - 4)
New: y - 1 = 1/2(x - 1)

Practice problems

Practice is necessary to master the point-slope form. Solve these problems to gain more understanding:

Problem 1

Write the equation of a line in point-slope form that passes through the point (-3, 7) with a slope of -2.

Solution:
y – 7 = -2(x + 3)

Problem 2

A line passes through the points (2, 4) and (6, 10). Find its equation in point-slope form.

Solution:
First calculate the slope:
m = (10 – 4) / (6 – 2) = 6 / 4 = 3/2

Now, use the point (2, 4):
y – 4 = 3/2(x – 2)

Conclusion

Through this exploration of the point-slope form, you have learned that this form is a powerful tool for determining the equation of a line using a known point and slope. Remember, practice using different points and slopes to get comfortable converting and simplifying equations.


Grade 10 → 3.6.2


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Slope-Intercept Form
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Two-Point Form

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Point-Slope Form

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