Question

In a ΔABC, if `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))`, then a², b², c² are in ______.

पर्याय

• A.P.

• G.P.

• H.P.

• None of these

Answer 1

In a ΔABC, if `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))`, then a², b², c² are in A.P..

Explanation:

`sin"A"/sin "C" =(sin"A" cos"B" - cos"A" sin"B")/(sin"B" cos"C" - cos"B" sin"C")`

⇒ `"a"/"c" = ("a"cos"B" - "b"cos"A")/("b"cos"C" - "c"cos"B")`

⇒ ab cos C – ac cos B = ac cos B – be cos A

⇒ ab cos C + be cos A= 2ac cos B

⇒ `("a"^2 + "b"^2 - "c"^2)/2 + ("b"^2 + "c"^2 - "a"^2)/2 = ("c"^2 + "a"^2 - "b"^2)/1`

⇒ b² = c² + a² – b²

⇒ b² = `("c"^2 + "a"^2)/2`

⇒ a², b², c² are in A.P.