### Notes from Toppers

# 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)*