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To compare the ratios a^2/b^2 and determine the intersection of the given pairs of linear equations, let's start by calculating the ratios for each pair:
(i) 5x - 4y + 8 = 0 and 7x + 6y - 9 = 0
To compare a^2/b^2, we need to find the values of a and b for each equation.
For the first equation (5x - 4y + 8 = 0):
a = coefficient of x = 5
b = coefficient of y = -4
For the second equation (7x + 6y - 9 = 0):
a = coefficient of x = 7
b = coefficient of y = 6
Now, calculate the ratios:
(a^2 / b^2) for the first equation = (5^2 / (-4)^2) = 25 / 16
(a^2 / b^2) for the second equation = (7^2 / 6^2) = 49 / 36
Since the ratios (25 / 16) and (49 / 36) are not equal, the lines representing the two equations intersect at a point.
(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
For the first equation (9x + 3y + 12 = 0):
a = coefficient of x = 9
b = coefficient of y = 3
For the second equation (18x + 6y + 24 = 0):
a = coefficient of x = 18
b = coefficient of y = 6
Now, calculate the ratios:
(a^2 / b^2) for the first equation = (9^2 / 3^2) = 81 / 9 = 9
(a^2 / b^2) for the second equation = (18^2 / 6^2) = 324 / 36 = 9
Since the ratios (9) and (9) are equal, the lines representing the two equations are coincident (meaning they are the same line).
(iii) 6x - 3y + 10 = 0 and 2x - y + 9 = 0
For the first equation (6x - 3y + 10 = 0):
a = coefficient of x = 6
b = coefficient of y = -3
For the second equation (2x - y + 9 = 0):
a = coefficient of x = 2
b = coefficient of y = -1
Now, calculate the ratios:
(a^2 / b^2) for the first equation = (6^2 / (-3)^2) = 36 / 9 = 4
(a^2 / b^2) for the second equation = (2^2 / (-1)^2) = 4 / 1 = 4
Since the ratios (4) and (4) are equal, the lines representing the two equations are parallel (meaning they have the same slope but do not intersect).
In summary:
(i) The lines represented by 5x - 4y + 8 = 0 and 7x + 6y - 9 = 0 intersect at a point.
(ii) The lines represented by 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0 are coincident (the same line).
(iii) The lines represented by 6x - 3y + 10 = 0 and 2x - y + 9 = 0 are parallel.
Verified answer
Step-by-step explanation:
1. (I) The given linear equation are
⇒5x−4y+8=0....eq1
⇒a1 =5,b1 =−4,c1=8
⇒7x+6y−9=0...eq2
⇒a2=7,b2 =6,c2 =−9
⇒a1/a2=5/7
⇒b1/b2= −4/6
⇒c1/c2 =8/9
comparing
⇒a1/a2,b1/b2,c1/c2
a1/a2 not equal to b1/b2
Hence,the line represented by eq1 and eq2 intersect at a point line represented by eq1 and eq2 intersect at a point
(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
a₁ = 9, b₁ = 3, c₁ = 12
a₂ = 18, b₂ = 6, c₂ = 24
a₁/a₂ = 9/18 = 1/2...(1)
b₁/b₂ = 3/6 = 1/2...(2)
c₁/c₂ = 12/24 = 1/2...(3)
From (1), (2) and (3)
a₁/a₂ = b₁/b₂ = c₁/c₂= 1/2
Therefore, they are coincident lines.
(iii) 6x – 3y + 10 = 0 and 2x – y + 9 = 0
a₁ = 6, b₁ = - 3, c₁ = 10
a₂ = 2, b₂ = - 1, c₂ = 9
a₁/a₂ = 6/2 = 3...(1)
b₁/b₂ = - 3/(- 1 ) = 3...(2)
c₁/c₂ = 10/9...(3)
From (1), (2) and (3)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Therefore, they are parallel lines.