Straight Lines Class 11 Formulas Slope Intercept Form and Solved Examples
In Euclidean geometry, a straight line is a set of all points between and stretching afar two points. In most geometry, a line is an elementary object that does not possess any formal properties beyond length, its single dimension. The two properties of straight lines in geometry are that they have only one length, dimension, and they stretch out only in two directions eternally. The idea of a straight line was first introduced by ancient mathematicians for the purpose of representing straight objects with negligible width and depth.
Angles Formed By A Straight Line
A straight line forms a 180-degree angle when constructing an angle arc from one point to another. Lines or straight lines are an idealization of such objects, which are frequently described in terms of two points or using a single letter.
Introduction To Point
A point is actually the simplest geometrical figure. It is a location in space, not having any dimension, depth, length, width, volume, or thickness. Even so, when you have two points, if you join every point in between those two points, you get a straight line.
Points on a line are referred to as collinear (col = "together" and linear = “line” or "string"). Only two points are required to identify a line.
How To Draw A Straight Line?
A straight line is one of the simplest drawings to construct in geometry. With a sheet of plain paper, a pencil, and a straightedge, you can draw a line or a straight line easily:
Firstly, make 2 dots on the sheet, some distance from each other; these are points
Then, the straightedge to join the 2 points with a pencil line, and stretch out the line beyond both points
Make arrowheads at the ends of the line you construct
Know About The Direction of Straight Lines
Horizontal Straight Lines: Straight lines can be in a horizontal direction, meaning that they are moving left and right of the viewing spot, endlessly.
Vertical Straight Lines: Straight lines can be in a vertical direction, meaning that they are rising above and drowning below the viewing spot, forever.
Diagonal Straight Lines: Straight lines can be in a diagonal direction, which is to say that they are any angle besides horizontal or vertical.
Parallel Line: Straight lines can be single or in pairs. Pairs of straight lines can run parallel to one another. Distance between two parallel lines is such that they never get closer and are always further apart. They are represented with the symbol ∥.
Intersecting Straight Lines: Pairs of straight lines bisect each other at any angle. When two straight lines bisect at perpendicular distance of 90°, they are perpendicular, represented with the symbol ⊥.
What are Curves in Geometry?
A curve is an opposite of a straight line just as a straight line is not a curve. A curved line consists of points that are non-linear to the two given points. The curve moves in other directions from the straight line formed by connecting collinear points.
Solved Example On Straight Line
Example:
Evaluate the angle between the y-axis and the line connecting the points (3, –1) and (4, –2).
Solution:
The slope of the line connecting the points (3, –1) and (4, –2) is
m= -2-(-1)/4-3
= -2 + 1 = -1
Now, the inclination (θ ) of the line connecting the points (3, –1) and (4, – 2) is allocated as:-
tan θ = –1
⇒ θ = (90° + 45°) = 135°
Therefore, the angle between the y-axis and the line connecting the points (3, –1) and (4, –2) is 135°.
FAQs on Straight Lines in Class 11 Maths Complete Concept Guide
1. What is a straight line in coordinate geometry?
A straight line in coordinate geometry is a one-dimensional figure that extends infinitely in both directions and can be represented by a linear equation in two variables. In Class 11 Maths, a straight line is usually written in the form ax + by + c = 0, where a, b, and c are constants and a and b are not both zero. Key points:
- It has a constant slope.
- Its graph on the Cartesian plane is a straight path.
- It represents a linear relationship between x and y.
2. What is the slope of a straight line?
The slope of a straight line is a measure of its steepness and is defined as the ratio of change in y to change in x. The formula for slope is m = (y₂ − y₁)/(x₂ − x₁). Important points:
- If m > 0, the line rises from left to right.
- If m < 0, the line falls from left to right.
- If m = 0, the line is horizontal.
- If x₂ = x₁, the line is vertical and slope is undefined.
3. What is the slope-intercept form of a straight line?
The slope-intercept form of a straight line is y = mx + c, where m is the slope and c is the y-intercept. In this form:
- m represents the slope (inclination of the line).
- c represents the point where the line cuts the y-axis.
- Example: In y = 2x + 3, slope = 2 and y-intercept = 3.
4. What is the general equation of a straight line?
The general equation of a straight line is ax + by + c = 0, where a, b, and c are real constants. Features of this form:
- It represents all straight lines in a plane.
- If b ≠ 0, it can be converted to slope form: y = (-a/b)x - (c/b).
- If a = 0, the line is horizontal.
- If b = 0, the line is vertical.
5. How do you find the equation of a line passing through two points?
The equation of a line passing through two points is found using the two-point form: y − y₁ = [(y₂ − y₁)/(x₂ − x₁)](x − x₁). Steps:
- Find slope m = (y₂ − y₁)/(x₂ − x₁).
- Substitute one point (x₁, y₁) into y − y₁ = m(x − x₁).
- m = (6−2)/(3−1) = 4/2 = 2
- Equation: y − 2 = 2(x − 1)
6. What is the point-slope form of a straight line?
The point-slope form of a straight line is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is useful when:
- Slope is known.
- One point on the line is given.
7. What is the intercept form of a straight line?
The intercept form of a straight line is x/a + y/b = 1, where a and b are the x-intercept and y-intercept respectively. Key details:
- a is the point where the line cuts the x-axis.
- b is the point where the line cuts the y-axis.
- Example: If intercepts are 4 and 2, equation is x/4 + y/2 = 1.
8. How do you find the angle between two straight lines?
The angle between two straight lines with slopes m₁ and m₂ is given by tanθ = |(m₂ − m₁)/(1 + m₁m₂)|. Steps:
- Find slopes m₁ and m₂.
- Substitute into the formula.
- Find θ using inverse tangent.
9. What is the condition for two straight lines to be parallel or perpendicular?
Two straight lines are parallel if their slopes are equal and perpendicular if the product of their slopes is −1. Conditions:
- Parallel lines: m₁ = m₂
- Perpendicular lines: m₁m₂ = −1
10. How do you find the distance of a point from a straight line?
The distance of a point (x₁, y₁) from a straight line ax + by + c = 0 is given by |ax₁ + by₁ + c| / √(a² + b²). Steps:
- Substitute the point coordinates into ax + by + c.
- Take the absolute value.
- Divide by √(a² + b²).
