Question

cho a/b=c/d Chứng minh a^2/c^2 = 2a^2 + 3b^2 / 2c^2 + 3d^2

Answer 1

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

\(\Leftrightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)

Ta có: \(\dfrac{a^2}{c^2}=\left(\dfrac{bk}{dk}\right)^2=\dfrac{b^2}{d^2}\)

\(\dfrac{2a^2+3b^2}{2c^2+3d^2}=\dfrac{2b^2k^2+3b^2}{2d^2k^2+3d^2}=\dfrac{b^2\left(2k^2+3\right)}{d^2\left(2k^2+3\right)}=\dfrac{b^2}{d^2}\)

Do đó: \(\dfrac{a^2}{c^2}=\dfrac{2a^2+3b^2}{2c^2+3d^2}\)