Straight Line Formulas – HSC Math

Posted In: Mathematics
1. Cartesian and polar co-ordinates:
x = r cosθ
$\r = √{\x^2+\y^2}$
$\θ = \tan^{-1}\y/\x$
2. Distance formula for (x,y) (a,b)
Distance = $√{(\x-\a)^2+(\y-\b)^2}$
for the polar, $√{(\r_1^2+\r_2^2-2\r_1\r_2\cos(\θ_1-\θ_2)}$
3. Division law:
internal division, $({\m_1\x_2+\m_2\x_1}/{\m_1+\m_2},{\m_1\y_2+\m_2\y_1}/{\m_1+\m_2})$
external division, $({\m_1\x_2-\m_2\x_1}/{\m_1-\m_2},{\m_1\y_2-\m_2\y_1}/{\m_1-\m_2})$
4. Middle point: $({\x_1+\x_2}/2,{\y_1+y_2}/2)$
5. Centroid of triangle: $({\x_1+\x_2+\x_3}/3,{\y_1+y_2+y_3}/3)$
6. Equation of straight line:
i. y = mx + c
ii. y – y1 = m(x – x1)
iii. ${\x-\x_1}/{\x_1-\x_2}={\y-\y_1}/{\y_1-\y_2}$
iv. $\x/\a + \y/\b = 1$
v. xcosθ + ysinθ = P
7. Slope:
i. m = tanθ
ii. m = ${\Δ\y}/{\Δ\x}$
8. Between two lines:
Angle, tanθ = $±{\m_1-\m_2}/{1-\m_1\m_2}$
Perpendicularity condition: m1m2 = -1
Parallelism condition: m1 = m2
9. For a line, ax + by + c = 0
parallel line, ax + by + k = 0
perpendicular line, bx – ay + k = 0
10. Perpendicular distance between line ax + by + c = 0 and point (x1,y1) is,
${|\a\x_1+\b\y_1+\c|}/√{\a^2+\b^2}$
Perpendicular distance between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is,
${|\c_2-\c_1|}/√{\a^2+\b^2}$
11. Bisectors of the angle between lines ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0 are,
${\a_1\x+\b_1\y+\c_1}/√{\a^2+\b^2}=±{\a_2\x+\b_2\y+\c_2}/√{\a^2+\b^2}$

Post Tags: Geometry | HSC