The Total solution for NCERT class 6-12
Prove that the area of the parallelogram formed by the linesa1x + b1y + c1 = 0, a1x + b1y+ d1 = 0, a2x + b2y + c2 =0, a2x + b2y + d2 = 0 is sq. units.
Deduce the condition for these lines to form a rhombus.
Answer - 1 : -
Given:
The given lines are
a1x + b1y+ c1 = 0 … (1)
a1x + b1y+ d1 = 0 … (2)
a2x + b2y+ c2 = 0 … (3)
a2x + b2y+ d2 = 0 … (4)
Let us prove, the areaof the parallelogram formed by the lines a1x + b1y + c1 =0, a1x + b1y + d1 = 0, a2x + b2y+ c2 = 0, a2x + b2y + d2 =0 is sq. units.
The area of theparallelogram formed by the lines a1x + b1y + c1 =0, a1x + b1y + d1 = 0, a2x + b2y+ c2 = 0 and a2x + b2y + d2 =0 is given below:
Hence proved.
Answer - 2 : -
3x − 4y + a= 0 … (1)
3x − 4y + 3a= 0 … (2)
4x − 3y − a= 0 … (3)
4x − 3y − 2a= 0 … (4)
Let us prove, the areaof the parallelogram formed by the lines 3x – 4y + a = 0, 3x – 4y + 3a = 0, 4x– 3y – a = 0 and 4x – 3y – 2a = 0 is 2a2/7 sq. units.
From above solution,we know that
Show that the diagonals of the parallelogram whose sides are lx + my + n= 0, lx + my + n’ = 0, mx + ly + n = 0 and mx + ly + n’ = 0 include an angleπ/2.
Answer - 3 : -
lx + my + n = 0 … (1)
mx + ly + n’ = 0 … (2)
lx + my + n’ = 0 … (3)
mx + ly + n = 0 … (4)
Let us prove, the diagonalsof the parallelogram whose sides are lx + my + n = 0, lx + my + n’ = 0, mx + ly+ n = 0 and mx + ly + n’ = 0 include an angle π/2.
By solving (1) and(2), we get
∴ m1m2 =-1