Question

Cho: a² + b²/c² + d² = ab/cd

CMR:a/b = c/d

 

Answer 1

Ta có : \(\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}=\frac{2ab}{2cd}=\frac{a^2+b^2+2ab}{c^2+d^2+2cd}=\frac{\left[a+b\right]^2}{\left[c+d\right]^2}=\left[\frac{a+b}{c+d}\right]^2(1)\)

\(\frac{a^2+b^2}{c^2+d^2}=\frac{2ab}{2cd}=\frac{a^2+b^2-2ab}{c^2+d^2-2cd}=\frac{\left[a-b\right]^2}{\left[c-d\right]^2}=\left[\frac{a-b}{c-d}\right]^2(2)\)

Từ 1 và 2 suy ra : \(\left[\frac{a+b}{c+d}\right]^2=\left[\frac{a-b}{c-d}\right]^2\)

Trường hợp 1 : \(\frac{a+b}{c+d}=\frac{a-b}{c-d}=\frac{a+b+a-b}{c+d+c-d}=\frac{2a}{2c}=\frac{a}{c}(3)\)

.\(\frac{a+b}{c+d}=\frac{a-b}{c-d}=\frac{a+b-a+b}{c+d-c+d}=\frac{2b}{2d}=\frac{b}{d}(4)\)

Từ 3 và 4 suy ra \(\frac{a}{c}=\frac{b}{d}\)hay \(\frac{a}{b}=\frac{c}{d}\).