Straight Lines

JEE Concepts:

1. Distance Formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Remember: This is the basic formula used to calculate the distance between two points in a two-dimensional plane.

2. Section Formula:

$$\text{Point R dividing}\space \overline{\text{PQ in the ratio m : n}}$$ $$(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n})$$ Remember: Use this formula to find the coordinates of a point R that divides a line segment PQ into a given ratio.

3. Mid-Point Formula:

$$\text{Midpoint M of}\space \overline{\text{AB}}$$ $$M=(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$$

Remember: This formula is specifically used to find the midpoint of a line segment when the coordinates of its endpoints are known.

4. Slope of a Straight Line:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Remember: The slope represents the steepness or inclination of a line and is expressed as a number.

5. Angle Between Two Lines:

$$\theta = \tan^{-1}\left|\frac{m_2 - m_1}{1 + m_1m_2}\right|$$

Remember: This formula is used to find the measure of the angle formed by two intersecting or non-intersecting lines.

6. Parallel and Perpendicular Lines:

• Two lines with the same slope are parallel.
• Two lines with slopes that are negative reciprocals of each other are perpendicular.

7. Equations of a Line: There are four main forms of linear equations:

Slope intercept form: $$y = mx + c$$ Intercept form: $$ax + by = c$$ Two point form: $$(y - y1)/(x - x1) = (y2 - y1)/(x2 - x1)$$ General form: $$Ax + By + C = 0$$

Remember: Each form has a specific usage, and the choice depends on the given information or the required output.

CBSE Concepts:

1. Distance Between Two Points:

$$\text{Distance between}\space (x_1, y_1) \space \text{and}\space (x_2, y_2)$$ $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

2. Slope of a Straight Line:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$ where (x1, y1) and (x2, y2) are any two distinct points on the line.

3. Linear Equations: Linear equations can be represented in different forms:

Slope-intercept form: $$y = mx + c$$ Intercept form: $$ax + by = c$$ Two-point form: $$(y - y1)/(x - x1) = (y2 - y1)/(x2 - x1)$$ where (x1, y1) and (x2, y2) are two points on the line and m is the slope of the line.

4. Parallel and Perpendicular Lines:

• Two lines with the same slope are parallel.
• Two lines with slopes that are negative reciprocals of each other are perpendicular.

5. Angles Formed by Two Lines: The angle formed by two intersecting lines can be calculated using the formula: $$\theta = \tan^{-1}\left|\frac{m_2 - m_1}{1 + m_1m_2}\right|$$ where m1 and m2 are the slopes of the two lines.