Formula sheet: Straight Lines

Straight Lines16 min readLesson 1 of 6

1.1 Slope and inclination The inclination of a line is the angle it makes with the positive direction of the x-axis, measured anticlockwise, and it lies between 0 and 180 degrees. The slope, or gradient, is the tangent of the inclination. A horizontal line has slope 0, and a vertical line has inclination 90 degrees with a slope that is not defined. When two points on the line are known, the slope is the change in y divided by the change in x. The order of subtraction must be the same in the numerator and the denominator, otherwise the sign flips. For the points (x1, y1) and (x2, y2) with x1 not equal to x2, the slope is (y2 minus y1) by (x2 minus x1).

m = tan θ = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Slope of a line from its inclination or from two points

1.2 Parallel, perpendicular and angle between lines Two non-vertical lines are parallel exactly when their slopes are equal. They are perpendicular exactly when the product of their slopes is minus 1, which also handles a horizontal line crossed with a vertical line, a case where the second slope simply is not available. For a pair of lines that both have slopes, the acute angle between them has tangent equal to the absolute value of (m1 minus m2) divided by (1 plus m1m2). The absolute value keeps the answer acute. If the denominator is zero the lines are perpendicular and the angle is 90 degrees. For the obtuse angle, take 180 degrees minus the acute angle.

m_{1} = m_{2} (∥), m_{1} m_{2} = - 1 (⊥), tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}

Parallel and perpendicular conditions and the angle between two lines

1.3 Forms of the equation of a line A horizontal line at height a has equation y equal to a, and a vertical line at distance b from the y-axis has equation x equal to b. The point-slope form through (x1, y1) with slope m is y minus y1 equal to m times (x minus x1). The two-point form through two given points is the point-slope form with the slope computed from the points. The slope-intercept form is y equal to mx plus c, where c is the y-intercept. The intercept form is x by a plus y by b equal to 1, where a and b are the x-intercept and the y-intercept. Every line has a general equation of the form Ax plus By plus C equal to 0, with A and B not both zero. Its slope is minus A by B, its x-intercept is minus C by A and its y-intercept is minus C by B, provided the denominators are not zero.

y - y_{1} = m (x - x_{1}), y = m x + c, \frac{x}{a} + \frac{y}{b} = 1

Point-slope, slope-intercept and intercept forms

1.4 Distance formulas The distance of a point (x1, y1) from the line Ax plus By plus C equal to 0 is the absolute value of Ax1 plus By1 plus C, divided by the square root of A squared plus B squared. The absolute value is essential, because distance cannot be negative but the expression inside can be. The distance between two parallel lines Ax plus By plus C1 equal to 0 and Ax plus By plus C2 equal to 0 is the absolute value of C1 minus C2 divided by the square root of A squared plus B squared. The two lines must be written with the same coefficients of x and y before the formula is applied, so scale one equation first if needed.

d = \frac{∣ A x _{1} + B y _{1} + C ∣}{A ^{2} + B ^{2}}, d = \frac{∣ C _{1} - C _{2} ∣}{A ^{2} + B ^{2}}

Distance of a point from a line and between parallel lines

Exam

Write each line in the same standard form before comparing coefficients. A missing factor of the square root of A squared plus B squared is the most common lost mark in distance questions.

in hinglish

Formula sheet yaad rakhne layak hai. Slope m = tan θ = (y₂-y₁)/(x₂-x₁); horizontal line ka slope 0 aur vertical ka undefined. Parallel lines ke slopes barabar, perpendicular ke slopes ka product -1. Do lines ke beech ka acute angle tan θ = |(m₁-m₂)/(1+m₁m₂)|. Line ke forms: point-slope y-y₁ = m(x-x₁), slope-intercept y = mx + c, intercept form x/a + y/b = 1, aur general form Ax + By + C = 0 jiska slope -A/B hota hai. Distance of point se line tak |Ax₁+By₁+C|/\sqrt(A²+B²), aur parallel lines ke beech |C₁-C₂|/\sqrt(A²+B²). Distance formula lagane se pehle dono lines ke x aur y ke coefficients same karo, warna answer galat aata hai. Distance wale questions me line ko pehle general form me likho aur dono taraf ke coefficients match karo.