




Β

Examples
Example 1 (b)
Example 1 (c) Important
Example 1 (d)
Example 2 Important
Example 3 Important
Example 4
Example 5
Example 6 Important
Example 7
Example 8
Example 9 (i)
Example 9 (ii) Important
Example 10
Example 11
Example 12
Example 13 Important
Example 14 Important
Example 15 Important
Example 16 You are here
Example 17
Example 18 Important
Example 19
Example 20
Example 21 Important
Example 22 Important
Example 23
Example 24 Important
Example 25 Important
Examples
Last updated at Feb. 3, 2020 by Teachoo
Β
Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 β 0 are: (i) Parallel if π_1/π_1 = π2/π2 The given lines are a1x + b1y + c1 = 0 & a2 x + b2 y + c2 = 0 Let slope of line (1) be m1 & slope of line (2) be m2 If two lines are parallel, then their slopes are equal If line (1) & (2) are parallel , then m1 = m2 Finding m1 & m2 From (1) a1x + b1y + c1 = 0 b1y = βc1 β a1 x b1y = βa1 x βc1 y = ( βπ_1 π₯ β π_1)/π_1 y = ((βπ_1)/π_1 ) x β(π_1/π_1 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (1) = m1 = (βπ_1)/π_1 From (2) a2x + b2y + c2 = 0 b2y = βc2 β a2 x b2y = βa2 x βc2 y = ( βπ_2 π₯ β π_2)/π_2 y = ((βπ_2)/π_2 )x + (π_2/π_2 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (2) = m2 = (βπ_2)/π_2 Since line (1) & (2) are parallel. So, m1 = m2 (βπ_1)/π_1 = (βπ_2)/π_2 ( π_π)/π_π = π_π/π_π Hence proved Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 β 0 are: (ii) Perpendicular if a1a2 + b1b2 = 0 . If two lines are perpendicular, then product of their slope is equal to β1 Since line (1) & (2) are perpendicular β (Slope of line 1) Γ (Slope of line 2) = β1 m1 Γ m2 = β 1 ( βπ_1)/π_1 Γ ( βπ_2)/π_2 = β1 ( π_1)/π_1 Γ π_2/π_2 = β1 a1a2 = βb1b2 a1a2 + b1b2 = 0 Hence proved