2. Show that the straight lines y? - 4y + 3 = 0 and x² + 4xy + 4y2 + 5x + 10y + 4 = 0 form a
parallelogram and find
the lengths of its sides.
Answers
Given : straight lines y² - 4y + 3 = 0 and x² + 4xy + 4y² + 5x + 10y + 4 = 0 form a parallelogram
To Find : Prove
find the lengths of its sides.
Solution:
y² - 4y + 3 = 0
=> ( y - 3)(y - 1) = 0
=> y = 3 , y = 1
Parallel lines
x² + 4xy + 4y² + 5x + 10y + 4 = 0
(x + 2y)² + 5(x + 2y) + 4 = 0
=> (x + 2y)² +4(x + 2y) + 1(x + 2y) + 4 = 0
=> (x + 2y + 1)(x + 2y + 4) = 0
x + 2y + 1 = 0
x + 2y + 4 = 0
Parallel lines
Hence parallelogram
x + 2y + 1 = 0 and y = 3 , y = 1
=> x = - 7 , - 3
x + 2y + 4 = 0 and y = 3 , y = 1
=> x = -10 , - 6
Points are ( -7 , 3) (-10 , 3) , ( - 6 , 1) , ( -3 , 1) ,
Length of sides = √(-7 + 10)² + ( 3 - 3)² = 3
√(-7 + 3)² + ( 3 - 1)² = √20 = 2√5
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