Important Straight Line Formulas For JEE
Straight lines are fundamental elements in geometry, and their concepts hold immense significance in the Joint Entrance Examination (JEE). Aspiring engineers seeking admission to top engineering institutes must have a firm grasp of straight line formulas. In this article, we'll explore the essential straight line formulas that are crucial for JEE preparation.
Understanding the Role of Straight Lines in JEE:
Straight lines serve as the building blocks for various geometric and algebraic concepts. In JEE, they find applications in coordinate geometry, calculus, and physics. A thorough understanding of straight line formulas not only aids in solving geometric problems but also lays the foundation for more advanced mathematical concepts.
Essential Straight Line Formulas for JEE:
Here's a breakdown of the key formulas that will strengthen your grasp on straight lines:
1. Slope of a Line (m): The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by:
- m = (y₂ - y₁) / (x₂ - x₁)
2. Angle between Two Lines (θ): The angle θ between two lines with slopes m₁ and m₂ is given by:
- tan(θ) = |(m₂ - m₁) / (1 + m₁ * m₂)|
3. Equation of a Line (y = mx + c): The equation of a line with slope m and y-intercept c is:
- y = mx + c
4. Point-Slope Form: The equation of a line passing through point (x₁, y₁) with slope m is:
- y - y₁ = m(x - x₁)
5. Intercept Form: The equation of a line with x-intercept a and y-intercept b is:
- x / a + y / b = 1
6. Two-Point Form: The equation of a line passing through points (x₁, y₁) and (x₂, y₂) is:
- (y - y₁) / (y₂ - y₁) = (x - x₁) / (x₂ - x₁)
7. Perpendicular Lines:
Two lines with slopes m₁ and m₂ are perpendicular if m₁ * m₂ = -1.
8. Parallel Lines:
Two lines with slopes m₁ and m₂ are parallel if m₁ = m₂.
9. Distance of a Point from a Line:
Distance of point (x₁, y₁) from line Ax + By + C = 0 is given by:
d = |Ax₁ + By₁ + C| / √(A² + B²)
10. Equation of a Line Parallel to Ax + By + C = 0:
Ax + By + k = 0 (k is constant)
11. Equation of a Line Perpendicular to Ax + By + C = 0:
Bx - Ay + k = 0 (k is constant)
12. Condition for Three Points to be Collinear:
Area of triangle formed by (x₁, y₁), (x₂, y₂), (x₃, y₃) = 0
13. Midpoint of a Line:
Midpoint between (x₁, y₁) and (x₂, y₂):
((x₁ + x₂) / 2, (y₁ + y₂) / 2)
14. Slope of Parallel Lines:
Parallel lines have the same slope.
15. Slope of Perpendicular Lines:
Slope of perpendicular lines are negative reciprocals.
16. Length of Projection on Axes:
Length of projection on x-axis: |a| / √(1 + m²)
Length of projection on y-axis: |b| / √(1 + m²)
17. Area of Triangle Formed by Lines:
If three lines are represented by ax + by + c₁ = 0, ax + by + c₂ = 0, and ax + by + c₃ = 0, then the area of the triangle formed is:
Area = 0.5 * |(c₁ + c₂ + c₃) / (a² + b²)|
18. Condition for Parallel Lines in Terms of Coefficients:
Two lines are parallel if a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂
19. Area of Triangle Formed by Points:
If the points are (x₁, y₁), (x₂, y₂), and (x₃, y₃), then the area of the triangle formed is:
Area = 0.5 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
20. Equation of Perpendicular Bisector:
Equation of perpendicular bisector of line segment between (x₁, y₁) and (x₂, y₂):
x(x₁ + x₂) / 2 + y(y₁ + y₂) / 2 = (x₁² + y₁² - x₂² - y₂²) / 2
21. Distance between Two Parallel Lines:
Distance between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0:
d = |C₂ - C₁| / √(A² + B²)
22. Distance between Two Intersecting Lines:
Distance between intersecting lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0:
d = |C₂ - C₁| / √(A² + B²)
These formulas are essential tools for tackling straight line problems in JEE Main and Advanced. Practice and familiarity with these formulas will greatly aid your success in the examination.
Applying Straight Line Formulas in JEE:
Mastery over these formulas is crucial for solving diverse problems in JEE. Be it finding angles
between lines, determining parallel or perpendicular lines, or solving problems involving intersections and tangents, these formulas are your tools for success.
Practice Makes Perfect:
Understanding the formulas is just the beginning. Regular practice is essential to internalize and apply them effectively. Solve a variety of problems from previous years' papers, sample papers, and textbooks to enhance your problem-solving skills.
Complementary Concepts:
While straight line formulas are vital, they are often intertwined with other geometric and algebraic concepts. Strengthen your foundation in coordinate geometry, vectors, and calculus, as they enhance your overall understanding and problem-solving capabilities.
Straight line formulas are not just mathematical expressions; they are the keys to unlocking the complexities of geometry and algebra in JEE. By mastering these formulas, you equip yourself with essential tools to approach problems confidently and efficiently. Invest time in understanding, practicing, and applying these formulas – they are the stepping stones towards your dream of excelling in the JEE and embarking on a successful engineering journey.
Important Straight Line Formulas For JEE FAQs
Q1: What is the formula for the straight line in JEE mains?
Ans: The formula for a straight line in JEE Main involves various concepts, including slope, intercepts, and equations. One of the basic formulas is the equation of a straight line in slope-intercept form: y = mx + c, where 'm' is the slope and 'c' is the y-intercept.
Q2: What is the important formula of a straight line?
Ans: One of the important formulas of a straight line is the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line, and 'm' is the slope. Additionally, the angle between two lines formula, and equations of lines in various forms (intercept form, two-point form) are also crucial.
Q3: What are the formulas of straight lines Class 12?
Ans: In Class 12, you'll encounter several formulas related to straight lines, including the slope of a line, point-slope form, two-point form, intercept form, angle between lines, distance of a point from a line, equations of lines in various forms, and more. These formulas are fundamental in coordinate geometry.
Q4: What is the straight line concept of JEE?
Ans: In the context of JEE, the straight line concept pertains to the study of lines in the coordinate plane. It involves understanding concepts like slope, equations of lines, angle between lines, distance of a point from a line, and more. These concepts are used to solve geometry problems and are a crucial part of the JEE syllabus.
Q5: What is the lowest mark in JEE?
Ans: The lowest mark in JEE can vary from year to year and depends on factors such as the difficulty level of the exam and the performance of the candidates. There is no fixed lowest mark as it is determined by the overall performance of all candidates who take the exam.
