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If the system of linear equations
2x + 2ay + az = 0
2x + 3by + bz = 0 and
2x + 4cy + cz = 0,
where a, b, c Є R are non-zero and distinct; has non-zero solution, then
- a)1/a, 1/b, 1/c are in A.P.
- b)a + b + c = 0
- c)a, b, c are in A.P.
- d)a, b, c are in G.P.
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If the system of linear equations2x + 2ay + az = 02x + 3by + bz = 0 an...
Given system of linear equations
2x + 2ay + az = 0
2x + 3by + bz = 0 and
2x + 4cy + cz = 0,
Now,
=> (3b − 2a)(c − a) − (4c − 2a)(b − a) = 0
=> 3bc − 2ac − 3ab + 2a2 − [4bc − 4ac − 2ab + 2a2] = 0
=> −bc + 2ac − ab = 0
=> ab + bc = 2ac
=> 1/c + 1/a = 2/b
Which shows that 1/a, 1/b, 1/c are in A.P.
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Community Answer
If the system of linear equations2x + 2ay + az = 02x + 3by + bz = 0 an...
To solve this problem, let's first rewrite the system of linear equations in matrix form:
⎡ 2 2a a ⎤ ⎡ x ⎤ ⎡ 0 ⎤
⎢ 2 3b b ⎥ ⋅ ⎢ y ⎥ = ⎢ 0 ⎥
⎣ 2 4c c ⎦ ⎣ z ⎦ ⎣ 0 ⎦
Next, let's find the determinant of the coefficient matrix:
| 2 2a a |
| 2 3b b |
| 2 4c c |
Expanding the determinant along the first row, we get:
2(3b - 4c) - 2a(2b - 2c) + a(2b - 3c)
= 6b - 8c - 4ab + 4ac + 2ab - 3ac
= -2ab + ac + 6b - 3c - 8c
Since the determinant is equal to zero for the system to have a non-zero solution, we have:
-2ab + ac + 6b - 3c - 8c = 0
-2ab + ac + 6b - 11c = 0
Now, let's simplify this equation:
a(c - 2b) + 6(b - 2c) = 0
a(c - 2b) = -6(b - 2c)
a = -6(b - 2c)/(c - 2b)
Since a, b, and c are non-zero and distinct, we can see that a, b, and c are in an arithmetic progression (A.P.) with common difference -6/(c - 2b).
Therefore, the correct answer is option A: 1/a, 1/b, 1/c are in an arithmetic progression (A.P.).
⎡ 2 2a a ⎤ ⎡ x ⎤ ⎡ 0 ⎤
⎢ 2 3b b ⎥ ⋅ ⎢ y ⎥ = ⎢ 0 ⎥
⎣ 2 4c c ⎦ ⎣ z ⎦ ⎣ 0 ⎦
Next, let's find the determinant of the coefficient matrix:
| 2 2a a |
| 2 3b b |
| 2 4c c |
Expanding the determinant along the first row, we get:
2(3b - 4c) - 2a(2b - 2c) + a(2b - 3c)
= 6b - 8c - 4ab + 4ac + 2ab - 3ac
= -2ab + ac + 6b - 3c - 8c
Since the determinant is equal to zero for the system to have a non-zero solution, we have:
-2ab + ac + 6b - 3c - 8c = 0
-2ab + ac + 6b - 11c = 0
Now, let's simplify this equation:
a(c - 2b) + 6(b - 2c) = 0
a(c - 2b) = -6(b - 2c)
a = -6(b - 2c)/(c - 2b)
Since a, b, and c are non-zero and distinct, we can see that a, b, and c are in an arithmetic progression (A.P.) with common difference -6/(c - 2b).
Therefore, the correct answer is option A: 1/a, 1/b, 1/c are in an arithmetic progression (A.P.).
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If the system of linear equations2x + 2ay + az = 02x + 3by + bz = 0 and2x + 4cy + cz = 0,where a, b, c R are non-zero and distinct; has non-zero solution, thena)1/a, 1/b, 1/c are in A.P.b)a + b + c = 0c)a, b, c are in A.P.d)a, b, c are in G.P.Correct answer is option 'A'. Can you explain this answer?
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