Notes for Straight Lines for the JEE Exam

1. Equation of a Straight Line

• Slope-intercept form: (y = mx + b)
• m: slope of the line
• b: y-intercept
• Point-slope form: (y - y_1 = m(x - x_1))
• $(x_1, y_1)$: a given point on the line
• m: slope of the line
• Two-point form: ((y - y_1)/(x - x_1) = m)
• $(x_1, y_1)$ and ((x_2, y_2))*: two given points on the line
• m: slope of the line

2. Parallel and Perpendicular Lines

• Parallel lines: lines with the same slope
• Perpendicular lines: lines with slopes that are negative reciprocals of each other

3. Distance Formula

• Distance between two points: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
• $(x_1, y_1)$ and ((x_2, y_2))*: two given points
• Distance from a point to a line:
• Distance between a given point and its orthogonal projection onto the line.

4. Intersection of Lines

• Finding the point of intersection of two lines:
  • Solve the system of equations formed by the equations of the two lines.
  • Determine whether the lines are parallel, perpendicular, or intersecting based on their slopes.

5. Angles Between Lines

• Finding the angle between two lines:
  • Calculate the slopes of the two lines.
  • Use the formula (\tan\theta = \frac{|m_2 - m_1|}{1 + m_1 m_2}), where m1 and m2 are the slopes of the two lines, and (\theta) is the angle between them.

6. Area of Triangles

• Using the coordinates of the vertices to find the area of a triangle:
• Calculate the area of the triangle using the formula (A = \frac{1}{2} \vert (x_1 y_2 - x_2 y_1) + (x_2 y_3 - x_3 y_2) + (x_3 y_1 - x_1 y_3) \vert), where A is the area, and x1, y1, x2, y2, x3, y3 are the coordinates of the vertices of the triangle.
• Utilize cross products as a method to calculate areas of triangles.

7. Linear Programming

• Graphical methods to solve linear inequalities
• Represent linear inequalities as regions on a graph
• Determine feasible regions and optimal solutions

8. Applications of Straight Lines

• Real-life situations involving rates of change
  • Motion in a straight line
  • Projectile motion
  • Uniform acceleration
  • Proportional relationships

9. Conic Sections

• Parabolas:
• Characteristics: Opens upward/downward, vertex, focus, directrix
• Equations: (y = ax^2 + bx + c) (Vertical) or (x = ay^2 + by + c) (Horizontal)
• Hyperbolas:
• Characteristics: Two branches, conjugate axis, foci, asymptotes
• Equations: (x^2/a^2 - y^2/b^2 = 1) (Transverse) or (y^2/b^2 - x^2/a^2 = 1) (Conjugate)
• Ellipses:
• Characteristics: Center, foci, vertices, major/minor axes
• Equations: ((x-h)^2/a^2 + (y-k)^2/b^2 = 1) (Horizontal) or ((y-k)^2/a^2 + (x-h)^2/b^2 = 1) (Vertical)
• Circles:
• Characteristics: Center, radius
• Equations: ((x-h)^2 + (y-k)^2 = r^2) (With center ((h, k)))

References:

• NCERT Mathematics Textbook for Class 11 (Chapter 10: Straight Lines)
• NCERT Mathematics Textbook for Class 12 (Chapter 6: Application of Derivatives)