Class 10 Exam > Class 10 Notes > Mathematics Class 10 ICSE > Revision Notes: Equation of A Line
Revision Notes: Equation of A Line | Mathematics Class 10 ICSE PDF Download
Important Concepts
- The angle which a straight line makes with the positive direction of x-axis measured in the anticlockwise direction is called the inclination (or angle of inclination) of the line. The inclination is usually denoted by θ.
- If θ is the inclination of a line l, then tan θ is called the slope or gradient of the line l.
- The slope of a line whose inclination is 90° is not defined.
- The slope of x-axis is zero and slope of y-axis is not defined.
- Three points A, B and C are collinear if Slope of AB = slope of BC.
- Let AB be a line cutting x-axis and the y-axis at A(a, 0) and B(0, b) respectively. Then the intercepts made on the axes are a and b respectively. That is, x-intercept = a and y-intercept = b.
- The equation of line parallel to x-axis at a distance a in the positive direction of y-axis is y = a and in negative direction of y-axis is y = -a.
Slope of a line
The slope m of the line through the points (x1, y1) and (x2, y2) is given by m =
Parallel and Perpendicular Lines
- Two non-vertical lines l and m are parallel if and only if their slopes are equal.
That is m1 = m2 - Two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.
That is m2 = (-1)/m1 ⇒ m1 x m2 = -1
Condition for Collinearity of Points
Three points A, B and C are collinear if Slope of AB = slope of BC
Equation of coordinate axes
- The equation of x-axis is y = 0
- The equation of y-axis is x = 0
Various Forms of the Equations of Straight Lines
- Slope-intercept form: The equation of a line having slope m and y-intercept c is given by y = mx + c.
- Point-Slope form: The equation of a line passing through (x1, y1) and having slope m is given by
y - y1 = m(x - x1). - Two-point form: The equation of line passing through two points A(x1, y1) and B(x2, y2) is given by
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FAQs on Revision Notes: Equation of A Line - Mathematics Class 10 ICSE
| 1. What is the general form of the equation of a line? |  |
Ans. The general form of the equation of a line in a two-dimensional Cartesian coordinate system is given by \( Ax + By + C = 0 \), where \( A \), \( B \), and \( C \) are constants, and \( A \) and \( B \) cannot both be zero.
| 2. How do you find the slope of a line given two points? | |
Ans. The slope \( m \) of a line that passes through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This gives the rate of change of \( y \) with respect to \( x \).
| 3. What is the slope-intercept form of a line, and how is it used? | |
Ans. The slope-intercept form of a line is expressed as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept (the point where the line crosses the y-axis). This form is useful for quickly graphing a line and understanding its characteristics.
| 4. How can you convert the general form of a line equation to slope-intercept form? | |
Ans. To convert the general form \( Ax + By + C = 0 \) to slope-intercept form \( y = mx + b \), you isolate \( y \) on one side of the equation. Rearranging gives \( By = -Ax - C \) and then dividing by \( B \) results in \( y = -\frac{A}{B}x - \frac{C}{B} \). Here, \( m = -\frac{A}{B} \) and \( b = -\frac{C}{B} \).
| 5. What are the different ways to represent the equation of a line? | |
Ans. The equation of a line can be represented in several forms: the general form \( Ax + By + C = 0 \), the slope-intercept form \( y = mx + b \), the point-slope form \( y - y_1 = m(x - x_1) \) (where \( (x_1, y_1) \) is a point on the line), and in parametric form. Each representation is useful in different mathematical contexts.
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