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a) áp dụng tính chất của dãy tỉ số bằng nhau ta có \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2-c^2}{b^2-d^2}\)
Do \(\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{ab}{cd}\)=> đpcm
b) áp dụng tính chất của dãy tỉ số bằng nhau ta có \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\Rightarrow\frac{ab}{cd}=\left(\frac{a-c}{b-d}\right)^2\)=> đpcm
\(\frac{a}{b}=\frac{c}{d}\)\(\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}\)
\(\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2+b^2}{c^2+d^2}=\frac{a^2-b^2}{c^2-d^2}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\left(\frac{a+b}{c+d}\right)^2=\frac{ab}{cd}\)
Vậy \(\frac{ab}{cd}=\frac{a^2-b^2}{c^2-d^2}\)và \(\left(\frac{a+b}{c+d}\right)^2=\frac{a^2+b^2}{c^2+d^2}\)
Cách 1:Đặt \(\frac{a}{b}=\frac{c}{d}=k\)
\(\Rightarrow\hept{\begin{cases}a=bk\\c=dk\end{cases}}\)
Rồi thay vào hai vế mà chứng minh
Cách 2:\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}\)
\(\Rightarrow\left(\frac{a}{c}\right)^2=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\frac{a}{c}.\frac{a}{c}=\frac{a}{c}.\frac{b}{d}=\frac{ab}{cd}\)
\(\Rightarrow\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\frac{ab}{cd}\)
a) ta có: \(\frac{a}{b}=\frac{c}{d}=k\)
\(\Rightarrow\frac{a}{b}=k\Rightarrow a=bk\)
\(\frac{c}{d}=k\Rightarrow c=dk\)
thay vào \(\frac{a^2-b^2}{ab}=\frac{\left(bk^2\right)-b^2}{bkb}=\frac{bkbk-bb}{bkb}=\frac{bb\times\left(kk-1\right)}{bbk}=\frac{kk-1}{k}\)
\(\frac{c^2-d^2}{cd}=\frac{\left(dk^2\right)-d^2}{dkd}=\frac{dkdk-dd}{dkd}=\frac{dd\times\left(kk-1\right)}{ddk}=\frac{kk-1}{k}\)
\(\Rightarrow\frac{a^2-b^2}{ab}=\frac{c^2-d^2}{cd}\left(=\frac{kk-1}{k}\right)\)
b) ta có \(\frac{a}{b}=\frac{c}{d}=k\)
\(\Rightarrow\frac{a}{b}=k\Rightarrow a=bk\)
\(\Rightarrow\frac{c}{d}=k\Rightarrow c=dk\)
thay vào \(\frac{\left(a+b\right)^2}{a^2+b^2}=\frac{\left(bk+b\right)^2}{bkbk+bb}=\frac{b\left(k+1\right)\times b\left(k+1\right)}{bb\left(kk+1\right)}=\frac{bb\left(k+1\right)\left(k+1\right)}{bb\left(kk+1\right)}=\frac{\left(k+1\right)\left(k+1\right)}{kk+1}\)
\(\frac{\left(c+d\right)^2}{c^2+d^2}=\frac{\left(dk+d\right)^2}{dkdk+dd}=\frac{\left(d\left(k+1\right)\right)^2}{dd\left(kk+1\right)}=\frac{d\left(k+1\right)\times d\left(k+1\right)}{dd\left(kk+1\right)}=\frac{dd\left(k+1\right)\left(k+1\right)}{dd\left(kk+1\right)}=\frac{\left(k+1\right)\left(k+1\right)}{kk+1}\)
\(\Rightarrow\frac{\left(a+b\right)^2}{a^2+b^2}=\frac{\left(c+d\right)^2}{c^2+d^2}\left(=\frac{\left(k+1\right)\left(k+1\right)}{kk+1}\right)\)
(a² + b²) / (c² + d²) = ab/cd
<=> (a² + b²)cd = ab(c² + d²)
<=> a²cd + b²cd = abc² + abd²
<=> a²cd - abc² - abd² + b²cd = 0
<=> ac(ad - bc) - bd(ad - bc) = 0
<=> (ac - bd)(ad - bc) = 0
<=> ac - bd = 0 hoặc ad - bc = 0
<=> ac = bd hoặc ad = bc
<=> a/b = d/c hoặc a/b = c/d (đpcm)
Ta có : \(\frac{a}{b}=\frac{c}{d}=\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}\)
\(\Rightarrow\frac{a}{c}.\frac{b}{d}=\frac{a+b}{c+d}.\frac{a+b}{c+d}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)
\(\Rightarrow\frac{ab}{cd}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)
Đặt \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=bk;c=dk\)
\(\Rightarrow\hept{\begin{cases}\frac{ab}{cd}=\frac{bk.b}{dk.d}=\frac{b^2}{d^2}\\\frac{a^2-b^2}{c^2-d^2}=\frac{b^2k^2-b^2}{d^2k^2-d^2}=\frac{b^2\left(k^2-1\right)}{d^2\left(k^2-1\right)}=\frac{b^2}{d^2}\end{cases}}\)
\(\Rightarrow\frac{ab}{cd}=\frac{a^2-b^2}{c^2-d^2}\)
và \(\Rightarrow\hept{\begin{cases}\left(\frac{a+b}{c+d}\right)^2=\left(\frac{bk+b}{dk+d}\right)^2=\frac{b^2\left(k+1\right)^2}{d^2\left(k+1\right)^2}=\frac{b^2}{d^2}\\\frac{a^2+b^2}{c^2+d^2}=\frac{b^2k^2+b^2}{d^2k^2+d^2}=\frac{b^2\left(k^2+1\right)}{d^2\left(k^2+1\right)}=\frac{b^2}{d^2}\end{cases}}\)
\(\Rightarrow\left(\frac{a+b}{c+d}\right)^2=\frac{a^2+b^2}{c^2+d^2}\)
