1
Question
If a, b, c are sides of a triangle and ∣∣
∣
∣∣a2b2c2(a+1)2(b+1)2(c+1)2(a−1)2(b−1)2(c−1)2∣∣
∣
∣∣=0, then ABC is a/an
If a, b, c are sides of a triangle and ∣∣
∣
∣∣a2b2c2(a+1)2(b+1)2(c+1)2(a−1)2(b−1)2(c−1)2∣∣
∣
∣∣=0, then ABC is a/an
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Solution
The correct option is C isosceles triangle
Δ=∣∣
∣
∣∣a2b2c2(a+1)2(b+1)2(c+1)2(a−1)2(b−1)2(c−1)2∣∣
∣
∣∣
Applying R2→R2−R3
=4∣∣
∣
∣∣a2b2c2abc(a−1)2(b−1)2(c−1)2∣∣
∣
∣∣
Applying R3→R3−R1+2R2
∴Δ=4∣∣
∣∣a2b2c2abc111∣∣
∣∣=−4(a−b)(b−c)(c−a)=0
If a – b = 0 or b – c = 0 or c – a = 0
∴ a = b or b = c or c = a
∴ΔABC is an isosceles triangle.
Δ=∣∣ ∣ ∣∣a2b2c2(a+1)2(b+1)2(c+1)2(a−1)2(b−1)2(c−1)2∣∣ ∣ ∣∣
Applying R2→R2−R3
=4∣∣ ∣ ∣∣a2b2c2abc(a−1)2(b−1)2(c−1)2∣∣ ∣ ∣∣
Applying R3→R3−R1+2R2
∴Δ=4∣∣ ∣∣a2b2c2abc111∣∣ ∣∣=−4(a−b)(b−c)(c−a)=0
If a – b = 0 or b – c = 0 or c – a = 0
∴ a = b or b = c or c = a
∴ΔABC is an isosceles triangle.
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