KnowledgeBoat Logo
|

Mathematics

If a+3b+2c+6da−3b+2c−6d=a+3b−2c−6da−3b−2c+6d\dfrac{a + 3b + 2c + 6d}{a - 3b + 2c - 6d} = \dfrac{a + 3b - 2c - 6d}{a - 3b - 2c + 6d}, prove that ab=cd\dfrac{a}{b} = \dfrac{c}{d}.

Ratio Proportion

2 Likes

Answer

Given,

⇒a+3b+2c+6da−3b+2c−6d=a+3b−2c−6da−3b−2c+6d\Rightarrow \dfrac{a + 3b + 2c + 6d}{a - 3b + 2c - 6d} = \dfrac{a + 3b - 2c - 6d}{a - 3b - 2c + 6d}

Applying componendo and dividendo, we get :

⇒(a+3b+2c+6d)+(a−3b+2c−6d)(a+3b+2c+6d)−(a−3b+2c−6d)=(a+3b−2c−6d)+(a−3b−2c+6d)(a+3b−2c−6d)−(a−3b−2c+6d)⇒(a+3b+2c+6d)+(a−3b+2c−6d)(a+3b+2c+6d)−a+3b−2c+6d=(a+3b−2c−6d)+(a−3b−2c+6d)(a+3b−2c−6d)−a+3b+2c−6d⇒2a+4c6b+12d=2a−4c6b−12d⇒2(a+2c)6(b+2d)=2(a−2c)6(b−2d)⇒a+2cb+2d=a−2cb−2d⇒a+2ca−2c=b+2db−2d\Rightarrow \dfrac{(a + 3b + 2c + 6d) + (a - 3b + 2c - 6d)}{(a + 3b + 2c + 6d) - (a - 3b + 2c - 6d)} = \dfrac{(a + 3b - 2c - 6d) + (a - 3b - 2c + 6d)}{(a + 3b - 2c - 6d) - (a - 3b - 2c + 6d)} \\[1em] \Rightarrow \dfrac{(a + 3b + 2c + 6d) + (a - 3b + 2c - 6d)}{(a + 3b + 2c + 6d) - a + 3b - 2c + 6d} = \dfrac{(a + 3b - 2c - 6d) + (a - 3b - 2c + 6d)}{(a + 3b - 2c - 6d) - a + 3b + 2c - 6d} \\[1em] \Rightarrow \dfrac{2a + 4c}{6b + 12d} = \dfrac{2a - 4c}{6b - 12d} \\[1em] \Rightarrow \dfrac{2(a + 2c)}{6(b + 2d)} = \dfrac{2(a - 2c)}{6(b - 2d)} \\[1em] \Rightarrow \dfrac{a + 2c}{b + 2d} = \dfrac{a - 2c}{b - 2d} \\[1em] \Rightarrow \dfrac{a + 2c}{a - 2c} = \dfrac{b + 2d}{b - 2d}

Applying componendo and dividendo again,

⇒(a+2c)+(a−2c)(a+2c)−(a−2c)=(b+2d)+(b−2d)(b+2d)−(b−2d)⇒(a+2c)+(a−2c)(a+2c)−a+2c=(b+2d)+(b−2d)(b+2d)−b+2d⇒2a4c=2b4d⇒ac=bd⇒ab=cd.\Rightarrow \dfrac{(a + 2c) + (a - 2c)}{(a + 2c) - (a - 2c)} = \dfrac{(b + 2d) + (b - 2d)}{(b + 2d) - (b - 2d)} \\[1em] \Rightarrow \dfrac{(a + 2c) + (a - 2c)}{(a + 2c) - a + 2c} = \dfrac{(b + 2d) + (b - 2d)}{(b + 2d) - b + 2d} \\[1em] \Rightarrow \dfrac{2a}{4c} = \dfrac{2b}{4d} \\[1em] \Rightarrow \dfrac{a}{c} = \dfrac{b}{d} \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}.

Hence, proved that ab=cd\dfrac{a}{b} = \dfrac{c}{d}.

Answered By

1 Like

Related Questions