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Ta có:\(\frac{3a+b+c+d}{a}=\frac{a+3b+c+d}{b}=\frac{a+b+3c+d}{c}=\frac{a+b+c+3d}{d}\)
\(\Rightarrow\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}\)
\(\Rightarrow\orbr{\begin{cases}a+b+c+d=0\\a=b=c=d\end{cases}}\)
\(TH1:a+b+c+d=0\Rightarrow\hept{\begin{cases}a+b=-\left(c+d\right)\\b+c=-\left(a+d\right)\end{cases}}\)
\(\Rightarrow Q=\left(\frac{-\left(c+d\right)}{c+d}\right)^2+\left(\frac{-\left(a+d\right)}{a+d}\right)^2+\left(\frac{c+d}{-\left(c+d\right)}\right)^2+\left(\frac{a+d}{-\left(a+d\right)}\right)^2\)
\(\Rightarrow Q=\left(-1\right)^2\cdot4=1\cdot4=4\)
\(TH2:a=b=c=d\)
\(\Rightarrow Q=\left(\frac{a+a}{a+a}\right)^2+\left(\frac{a+a}{a+a}\right)^2+\left(\frac{a+a}{a+a}\right)^2+\left(\frac{a+a}{a+a}\right)^2=1^2\cdot4=1\cdot4=4\)
Vậy Q=4
Gọi \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=kb;c=kd\)(1)
Thay (1) vào ta có :
\(\frac{3a^2+c^2}{3b^2+d^2}=\frac{3\left(kb\right)^2+\left(kd\right)^2}{3b^2+d^2}=\frac{3k^2b^2+k^2+d^2}{3b^2+d^2}=\frac{k^2\left(3b^2+d^2\right)}{3b^2+d^2}=k^2\)(1)
\(\frac{\left(a+c\right)^2}{\left(b+d\right)^2}=\frac{\left(kb+kd\right)^2}{\left(b+d\right)^2}=\frac{\left[k\left(b+d\right)\right]^2}{\left(b+d\right)^2}=\frac{k^2.\left(b+d\right)^2}{\left(b+d\right)^2}=k^2\)(2)
Từ (1) và (2)
\(\Rightarrow\frac{3a^2+c^2}{3b^2+d^2}=\frac{\left(a+c\right)^2}{\left(b+d\right)^2}\)
\(\RightarrowĐPCM\)
Đặt \(\frac{a}{b}=\frac{c}{d}=k\)\(\Rightarrow a=bk;c=dk\)
1)Xét \(VT=\frac{\left(bk\right)^2+bkdk}{\left(dk\right)^2-bkdk}=\frac{b^2k^2+bdk^2}{d^2k^2-bdk^2}=\frac{k^2\left(b^2+bd\right)}{k^2\left(d^2-bd\right)}=\frac{b^2+bd}{d^2-bd}=VP\)
Suy ra Đpcm
2)Xét \(VT=\frac{3\left(bk\right)^2+\left(dk\right)^2}{3b^2+d^2}=\frac{3b^2k^2+d^2k^2}{3b^2+d^2}=\frac{k^2\left(3b^2+d^2\right)}{3b^2+d^2}=k^2\left(1\right)\)
Xét \(VP=\frac{\left(bk+dk\right)^2}{\left(b+d\right)^2}=\frac{k^2\left(b+d\right)^2}{\left(b+d\right)^2}=k^2\left(2\right)\)
Từ (1) và (2) suy ra Đpcm
Đăt \(\frac{a}{b}=\frac{c}{d}=k\)\(\Rightarrow a=bk;c=dk\)
Khi đó \(\frac{3a^2+c^2}{3b^2+d^2}=\frac{3.\left(bk\right)^2+\left(dk^2\right)}{3.b^2+d^2}=\frac{3b^2.k^2+d^2.k^2}{3b^2+d^2}=\frac{k^2.\left(3b^2+d^2\right)}{3b^2+d^2}=k^2\) (1)
\(\frac{\left(a+c\right)^2}{\left(b+d\right)^2}=\frac{\left(bk+dk\right)^2}{\left(b+d\right)^2}=\frac{k^2.\left(b+d\right)^2}{\left(b+d\right)^2}=k^2\)(2)
Từ (1) và (2) ta có \(\frac{3a^2+c^2}{3b^2+d^2}=\frac{\left(a+c\right)^2}{\left(b+d\right)^2}\)
Ta có: a/b = c/d => a/b.c/d = c/d.c/d (vì các p/s nào bằng nhau nhân với mấy cũng bằng nhau)
hay: ac/d = c^2/d^2 (1)
Lại có: a/b = c/d = a^2/b^2 = c^2/d^2 = a^2+c^2/b^2+d^2 (2)
Từ (1) và (2) => ac/bd = a^2+c^2/b^2/d^2
a, Ta có: \(\frac{a}{b}=\frac{c}{d}=k\left(k\ne0\right)\Rightarrow a=kb;c=kd\)
Thay:
\(\frac{ab}{cd}=\frac{b^2}{d^2}\)
\(\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\frac{b^2\left(k+1\right)^2}{d^2\left(k+1\right)^2}=\frac{b^2}{d^2}\)
=> đpcm
a) \(\frac{a}{b}=\frac{c}{d}=\frac{11a}{11b}=\frac{9c}{9d}\)
áp dụng t/c dãy tỉ số bằng nhau ta có:
\(\frac{a}{b}=\frac{c}{d}=\frac{11a+9c}{11b+9d}\)
\(\Rightarrow\frac{a}{b}=\frac{11a+9c}{11c+9d}\left(đpcm\right)\)
b) \(\frac{a}{b}=\frac{c}{d}=\frac{3a}{3b}=\frac{5c}{5d}\Rightarrow\frac{a^2}{b^2}=\frac{c^2}{d^2}=\frac{3a^2}{3b^2}=\frac{5c^2}{5d^2}\)
áp dụng t/c dãy tỉ số bằng nhau ta có:
\(\frac{a^2}{b^2}=\frac{c^2}{d^2}=\frac{3a^2+5c^2}{3b^2+5d^2}\left(1\right)\)
\(\frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+d}\)
\(\Rightarrow\frac{a^2}{b^2}=\frac{\left(a+c\right)^2}{\left(b+d\right)^2}\left(2\right)\)
từ (1) và (2) => \(\frac{3a^2+5c^2}{3b^2+5d^2}=\frac{\left(a+c\right)^2}{\left(b+d\right)^2}\left(đpcm\right)\)
Dễ mà
Ta có: \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\)
Áp dụng t/c dãy tỉ số bằng nhau:
Ta có: \(\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}=\frac{a-b}{c-d}\)(1)
Từ (1),
Ta có: \(\frac{a+b}{c+d}\cdot\frac{a+b}{c+d}=\frac{a+b}{c+d}\cdot\frac{a-b}{c-d}\)(nhân mỗi vế với \(\frac{a+b}{c+d}\))
Vậy \(\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\frac{\left(a+b\right)\left(a-b\right)}{\left(c+d\right)\left(c-d\right)}=\frac{a^2-b^2}{c^2-d^2}\)(đpcm)