In a ∆ABC, if sin A/sin B = sin(A-b)/sin(A-c) and a, b, c are the sides of the ∆ABC, then the correct relationship between a, b, c is ___.
(a) c2 (a2 + b2) = a2 (c2 - b2)
(b) a2 (b2 - c2) = b2 (a2 - c2)
(c) ac(a2 - c2) = b2 (a2 - b2)
(d) bc(a2 - c2) = a2 (b2 - c2)
Correct answer: (c) ac(a2 - c2) = b2 (a2 - b2)
Explanation:
\(\frac{sin\ A}{sin \ B}\) = \(\frac{sin(A -B)}{sin(A - C)}\)
⇒ \(\frac{a}{b}\) = \(\frac{sin A.cos\ B - cosA.sinB}{sinA.cosC - cosA.sinC}\)
⇒ \(\frac{a}{b}\) = \(\frac{a cos \ B - b. cos \ A}{a.cos \ C - c.cos \ A}\)
⇒ \(\frac{a}{b}\) = \(\frac{a.cos\ B-(c-acos\ B)}{acos\ C-(b-acos \ C)}\)
⇒ \(\frac{a}{b}\) = \(\frac{cos \ B(2a)-c}{cos \ C(2a)-b}\)
⇒ \(\frac{a}{b}\) = \(\frac{(\frac{c^2 + a^2 -b^2}{c})-c}{(\frac{b^2 + a^2 - c^2}{b})-b}\)
⇒ \(\frac{a}{b}\) = \(\frac{b(a^2 - b^2)}{c(a^2 - c^2)}\)
⇒ ac(a2 - c2) = b2 (a2 - b2)