8a−5b8c−5d=8a+5b8c+5d, prove that ab=cd.\dfrac{8a - 5b}{8c - 5d} = \dfrac{8a + 5b}{8c + 5d}, \text{ prove that } \dfrac{a}{b} = \dfrac{c}{d}.8c−5d8a−5b=8c+5d8a+5b, prove that ba=dc.
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Given,
8a−5b8c−5d=8a+5b8c+5d\dfrac{8a - 5b}{8c - 5d} = \dfrac{8a + 5b}{8c + 5d} \\[0.5em]8c−5d8a−5b=8c+5d8a+5b
By alternendo,
⇒8a−5b8a+5b=8c−5d8c+5d\Rightarrow \dfrac{8a - 5b}{8a + 5b} = \dfrac{8c - 5d}{8c + 5d} \\[0.5em]⇒8a+5b8a−5b=8c+5d8c−5d
By componendo & dividendo,
⇒8a−5b+8a+5b8a−5b−8a−5b=8c−5d+8c+5d8c−5d−8c−5d⇒−16a10b=−16c10d\Rightarrow \dfrac{8a - 5b + 8a + 5b}{8a - 5b - 8a - 5b} = \dfrac{8c - 5d + 8c + 5d}{8c - 5d - 8c - 5d} \\[1em] \Rightarrow -\dfrac{16a}{10b} = -\dfrac{16c}{10d}⇒8a−5b−8a−5b8a−5b+8a+5b=8c−5d−8c−5d8c−5d+8c+5d⇒−10b16a=−10d16c
On dividing the equation by −1610-\dfrac{16}{10}−1016,
⇒ab=cd\Rightarrow \dfrac{a}{b} = \dfrac{c}{d} \\[0.5em]⇒ba=dc
Hence, proved that ab=cd.\dfrac{a}{b} = \dfrac{c}{d}.ba=dc.
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If a : b : : c : d, prove that
(i)2a+5b2a−5b=2c+5d2c−5d.(ii)5a+11b5c+11d=5a−11b5c−11d.(iii)(2a+3b)(2c−3d)=(2a−3b)(2c+3d).(iv)(la+mb):(lc+md)::(la−mb):(lc−md).\begin{matrix} \text{(i)} & \dfrac{2a + 5b}{2a - 5b} = \dfrac{2c + 5d}{2c - 5d}. \\[0.5em] \text{(ii)} & \dfrac{5a + 11b}{5c + 11d} = \dfrac{5a - 11b}{5c - 11d}. \\[0.5em] \text{(iii)} & (2a + 3b)(2c - 3d) \ & = (2a - 3b)(2c + 3d). \\ \text{(iv)} & (la + mb) : (lc + md) \ & : : (la - mb) : (lc - md). \end{matrix}(i)(ii)(iii)(iv)2a−5b2a+5b=2c−5d2c+5d.5c+11d5a+11b=5c−11d5a−11b.(2a+3b)(2c−3d)=(2a−3b)(2c+3d).(la+mb):(lc+md)::(la−mb):(lc−md).
If 5x+7y5u+7v=5x−7y5u−7v, show that xy=uv.\dfrac{5x + 7y}{5u + 7v} = \dfrac{5x - 7y}{5u - 7v}, \text{ show that } \dfrac{x}{y} = \dfrac{u}{v}.5u+7v5x+7y=5u−7v5x−7y, show that yx=vu.
If (4a + 5b)(4c - 5d) = (4a - 5b)(4c + 5d), prove that a, b, c, d are in proportion.
If (pa + qb) : (pc + qd) : : (pa - qb) : (pc - qd), prove that a : b : : c : d.
