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Question
Select the correct option from the given alternatives:
The determinant D = `|("a", "b", "a" + "b"),("b", "c", "b" + "c"),("a" + "b", "b" + "c", 0)|` = 0 if
Options
a, b, c are in A.P.
a, b, c are in G.P
a, b, c are in H.P.
α is root of ax2 + 2bx + c = 0
Solution
a, b, c are in G.P.
Explanation:
Applying R3 → R3 – (R1 + R2), we get
`|("a", "b", "a" + "b"),("b", "c", "b" + "c"),(0, 0, -("a" + 2"b" + "c"))|` = 0
∴ a[–c(a + 2b + c) – 0] – b[–b(a + 2b + c) – 0] + (a + b) (0 – 0) = 0
∴ (–ac + b2) (a + 2b + c) = 0
∴ –ac + b2 = 0 or a + 2b + c = 0
∴ b2 = ac
∴ a, b, c are in G.P.
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