Question

5) If a + b/b + c = c + d/d + a, p.t c = a or a + b + c + d = 0
Please help​

Answer 1

$\large\underline\blue{\bold{Given \:  }}$

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$\tt \:  \longrightarrow \: \dfrac{a + b}{b + c} = \dfrac{c + d}{d + a}$

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$\large\underline\blue{\bold{To \:  prove \: - }}$

$\tt \:  \longrightarrow \: a = c \: or \: a + b + c + d = 0$

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$\large\underline\purple{\bold{Solution :- }}$

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$\tt \:  \longrightarrow \: \dfrac{a + b}{b + c} = \dfrac{c + d}{d + a}$

$\tt \:  \longrightarrow \: (a + b)(d + a) = (b + c)(c + d)$

$\tt \:  \longrightarrow \: ad + {a}^{2} + bd + ba = bc + bd + {c}^{2} + cd$

$\tt \:  \longrightarrow \: ad + {a}^{2} + ab - bc - {c}^{2} - cd = 0$

$\tt \:  \longrightarrow \: ( {a}^{2} - {c}^{2} ) + b(a - c) + d(a - c) = 0$

$\tt \:  \longrightarrow \: (a - c)(a + c) + b(a - c) + d(a - c) = 0$

$\tt \:  \longrightarrow \: (a - c)(a + c + b + d) = 0$

$\tt\implies \:a - c = 0 \: or \: a + b + c + d = 0$

$\tt\implies \: \boxed{ \red{ \bf \:a = c \: or \: a + b + c + d = 0 }}$

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$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

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Answer 2

EXPLANATION.

⇒ a + b/b + c = c + d/d + a.

As we know that,

Take L.C.M on both sides, we get.

⇒ (a + b)(d + a) = (c + d)(b + c).

⇒ ad + a² + bd + ab = bc + c² + bd + cd.

⇒ ad + a² + bd + ab - bc - c² - bd - cd = 0.

⇒ ad + a² + ab - bc - c² - cd = 0.

⇒ a² - c² + ad - cd + ab - bc = 0.

⇒ (a - c)(a + c) + d(a - c) + b(a - c) = 0.

⇒ (a - c)(a + c + d + b) = 0.

⇒ a - c = 0.

⇒ a = c.

⇒ a + b + c + d = 0.