Question

if a:b = c:d show that xa + yb: : a b xc yd : c d α− β= + α− β

Answer 1

SOLUTION

TO PROVE

If a : b = c : d then show that

$\displaystyle \sf{(xa + yb) : (a \alpha - b \beta ) =(xc + yd) : (c \alpha - d \beta ) }$

EVALUATION

Here it is given that a : b = c : d

$\displaystyle \sf{ \implies \: \frac{a}{b} = \frac{c}{d} = k \: \: \: (say)}$

$\displaystyle \sf{ \implies \: a = bk \: \: and \: \: c = dk}$

Now

LHS

$\displaystyle \sf{ = (xa + yb) : (a \alpha - b \beta ) }$

$\displaystyle \sf{ = \frac{ (xa + yb)}{(a \alpha - b \beta )} }$

$\displaystyle \sf{ = \frac{ (xbk + yb)}{(bk\alpha - b \beta )} }$

$\displaystyle \sf{ = \frac{b (xk + y)}{b(k\alpha - \beta )} }$

$\displaystyle \sf{ = \frac{ (xk + y)}{(k\alpha - \beta )} }$

RHS

$\displaystyle \sf{=(xc + yd) : (c \alpha - d \beta ) }$

$\displaystyle \sf{= \frac{(xc + yd)}{(c \alpha - d \beta )} }$

$\displaystyle \sf{= \frac{(xdk + yd)}{(dk \alpha - d \beta )} }$

$\displaystyle \sf{= \frac{d(xk + y)}{d(k \alpha - \beta )} }$

$\displaystyle \sf{= \frac{(xk + y)}{(k \alpha - \beta )} }$

∴ LHS = RHS

Hence proved

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