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Examples

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Last updated at Feb. 3, 2020 by Teachoo

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