Question

In $ \Delta ABC $ , if $ \sin A:\sin C = \sin (A - B):\sin (B - C) $ , then $ {a^2} $ , $ {b^2} $ and $ {c^2} $ are in
A. A.P
B. G.P
C. H.P
D. None of these

Answer 1

Hint: In this question, a triangle is given which satisfy the condition $ \sin A:\sin C = \sin (A - B):\sin (B - C) $ , we have to find relation of $ {a^2} $ , $ {b^2} $ and $ {c^2} $ .
Use the property of a triangle, the sum of all angles of a triangle is $ {180^ \circ } $ , to write the angle in terms of the other two angles.
 $ \Rightarrow $ $ A + B + C = {180^ \circ } $
According to the trigonometrically ratios of angles \[\left( {180^\circ {\text{ }} - {\text{ }}\theta } \right)\] , \[\sin \left( {180^\circ {\text{ }} - {\text{ }}\theta } \right) = \sin \theta \] .
Now, use the trigonometric formula, \[\sin (A + B)\sin (A - B) = {\sin ^2}A - {\sin ^2}B\]
Apply the Sine rule of the triangle,
If a, b and c are the sides of the triangle and their corresponding opposite angles are A , B and C then,
 $ \dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}} = k $
From this relation write $ \sin A $ , $ \sin B $ and $ \sin C $ in terms of a, b, c and k.
If $ {a^2} $ , $ {b^2} $ and $ {c^2} $ are in A.P. then the difference between the consecutive terms is the same.
 $ \Rightarrow {b^2} - {a^2} = {c^2} - {b^2} $
 $ \Rightarrow 2{b^2} = {c^2} + {a^2} $

Complete step-by-step answer:
Consider a triangle , where A, B and C are the angles and a, b, and c are the sides.
Given is the equation, $ \sin A:\sin C = \sin (A - B):\sin (B - C) $ which is same as,
 $ \dfrac{{\sin A}}{{\sin C}} = \dfrac{{\sin (A - B)}}{{\sin (B - C)}} \ldots (1) $
Since the sum of all angles of a triangle is $ {180^ \circ } $ .
 $ A + B + C = {180^ \circ } $
 $ \Rightarrow A = {180^ \circ } - (B + C) \ldots (2) $
 And, find the angle C,
 $ C = {180^ \circ } - (A + B) \ldots (3) $
Substitute the value of A from equation $ (1) $ and C from equation $ (2) $ into the left-hand side of the equation $ (3) $ .
  $ \dfrac{{\sin A}}{{\sin C}} = \dfrac{{\sin (A - B)}}{{\sin (B - C)}} $
 $ \dfrac{{\sin ({{180}^ \circ } - (B + C))}}{{\sin ({{180}^ \circ } - (A + B))}} = \dfrac{{\sin (A - B)}}{{\sin (B - C)}} $
Apply the trigonometrically ratios of angles \[\left( {180^\circ {\text{ }} - {\text{ }}\theta } \right)\] , \[\sin \left( {180^\circ {\text{ }} - {\text{ }}\theta } \right) = \sin \theta \] .
 $ \therefore \dfrac{{\sin (B + C)}}{{\sin (A + B)}} = \dfrac{{\sin (A - B)}}{{\sin (B - C)}} $
After cross multiplying the equation we get,
 $ \Rightarrow \sin (B + C)\sin (B - C) = \sin (A - B)\sin (A + B) \ldots (4) $
Apply the trigonometric formula, \[\sin (A + B)\sin (A - B) = {\sin ^2}A - {\sin ^2}B\] to the equation $ (4) $ .
 $ \Rightarrow {\sin ^2}B - {\sin ^2}C = {\sin ^2}A - {\sin ^2}B $
 $ \Rightarrow 2{\sin ^2}B = {\sin ^2}A + {\sin ^2}C \ldots (5) $
Apply the Sine rule of the triangle,
If a, b and c are the sides of the triangle and their corresponding opposite angles are A , B and C then,
 $ \dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}} = k $
Take the ratio,
 $ \dfrac{b}{{\sin B}} = k $
 $ \Rightarrow \dfrac{b}{k} = \sin B $
Now, take the ratio,
 $ \dfrac{a}{{\sin A}} = k $
 $ \Rightarrow \dfrac{a}{k} = \sin A $
Now, take the ratio,
 $ \dfrac{c}{{\sin C}} = k $
 $ \Rightarrow \dfrac{c}{k} = \sin C $
Substitute the values of $ \sin A $ , $ \sin B $ and $ \sin C $ into the equation $ (5) $
 $ \Rightarrow 2{\left( {\dfrac{b}{k}} \right)^2} = {\left( {\dfrac{a}{k}} \right)^2} + {\left( {\dfrac{c}{k}} \right)^2} $
 $ \Rightarrow \dfrac{{2{b^2}}}{{{k^2}}} = \dfrac{{{a^2} + {c^2}}}{{{k^2}}} $
 $ \Rightarrow 2{b^2} = {a^2} + {c^2} $
If $ {a^2} $ , $ {b^2} $ and $ {c^2} $ are in A.P. then the difference between the consecutive terms is the same.
 $ \Rightarrow {b^2} - {a^2} = {c^2} - {b^2} $
 $ \Rightarrow 2{b^2} = {c^2} + {a^2} $

Correct Answer : A) A.P.

Note:
An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. That is if a, b ,c are in A.P. then, \[b - a = c - b\] . A sequence of numbers is called a geometric progression if the ratio of any two consecutive terms is same i.e. For example 2,4,8,16 is a GP because ratio of any two consecutive terms in the series is same , \[\dfrac{4}{2}{\text{ }} = \dfrac{8}{4} = \dfrac{{16}}{8} = 2\] . A sequence of numbers is called a harmonic progression if the reciprocal of the terms are in AP. In simple terms, a ,b, c, are in HP if $ \dfrac{1}{a} $ , $ \dfrac{1}{b} $ , $ \dfrac{1}{c} $ are in AP.