Given-a-b-c-d-b-a-d-c-Show-that-a-2-b-2-c-2-d-2-b-2-a-2-d-2-c-2-

Question Number 71242 by TawaTawa last updated on 13/Oct/19

Given : \frac{a}{b} + \frac{c}{d} = \frac{b}{a} + \frac{d}{c}

Show that, \frac{a^{2}}{b^{2}} - \frac{c^{2}}{d^{2}} = \frac{b^{2}}{a^{2}} - \frac{d^{2}}{c^{2}}

Answered by MJS last updated on 13/Oct/19

α + β = \frac{1}{α} + \frac{1}{β}

\Rightarrow β = - α \lor β = \frac{1}{α}

α^{2} - β^{2} = \frac{1}{α^{2}} - \frac{1}{β^{2}} {\begin{cases} β = - α \Rightarrow true for α \neq 0 \\ β = \frac{1}{α} \Rightarrow true only for α = \pm 1 \end{cases}

\Rightarrow question wrong ? α^{2} - β^{2} = \frac{1}{β^{2}} - \frac{1}{α^{2}} is always

true for α \neq 0 \land (β = - α \lor β = \frac{1}{α})

Commented by $@ty@m123 last updated on 13/Oct/19

T h a n k s f o r t h i s n e w (f o r m e) c o n c e p t .

B u t t h e n w h e r e i s t h e m i s t a k e

i n m y s o u t i o n .

I a m r e p e a t e d l y c h e k i n g

b u t f a i l e d t o f i n d .

Commented by MJS last updated on 13/Oct/19

you have made no mistake and your solution

is the same as mine : the question is “ {wrong}^{″}

Commented by TawaTawa last updated on 14/Oct/19

God bless you sir

Answered by JDamian last updated on 13/Oct/19

i f c t u r n s i n t o - c, t h e n

\frac{a}{b} - \frac{c}{d} = \frac{b}{a} - \frac{d}{c}

(\frac{a}{b} + \frac{c}{d}) \times (\frac{a}{b} - \frac{c}{d}) = (\frac{b}{a} + \frac{d}{c}) \times (\frac{b}{a} - \frac{d}{c})

\frac{a^{2}}{b^{2}} - \frac{c^{2}}{d^{2}} = \frac{b^{2}}{a^{2}} - \frac{d^{2}}{c^{2}}

Commented by $@ty@m123 last updated on 13/Oct/19

I h a v e a d o u b t o n t h i s p r o c e s s .

I f i t w o u l d b e s o m e t h i n g l i k e :

a + b = c

W o u l d y o u s t i l l w r i t e

“ i f c t u r n s i n t o - c, {t h e n}^{″}

Commented by TawaTawa last updated on 14/Oct/19

God bless you sir

Answered by $@ty@m123 last updated on 13/Oct/19

S o l u t i o n :

Given : \frac{a}{b} + \frac{c}{d} = \frac{b}{a} + \frac{d}{c}

\Rightarrow \frac{a d + b c}{b d} = \frac{b c + a d}{a c}

\Rightarrow b d = a c \dots . . (1)

N o w,

L H S = \frac{a^{2}}{b^{2}} - \frac{c^{2}}{d^{2}}

= \frac{a^{2} d^{2} - b^{2} c^{2}}{b^{2} d^{2}}

= \frac{a^{2} d^{2} - b^{2} c^{2}}{a^{2} c^{2}}

= \frac{d^{2}}{c^{2}} - \frac{b^{2}}{a^{2}}

= R H S

p l . c h e c k y o u r q u e s t i o n .

Commented by TawaTawa last updated on 14/Oct/19

God bless you sir