回復 2# bombwemg 的帖子
第5題
設\(x,y\)均大於0,則聯立方程組\(\cases{\displaystyle \frac{4x+3y-6}{4x+3y}=\frac{2}{\sqrt{x}}\cr \frac{4x+3y+6}{4x+3y}=\frac{4}{\sqrt{y}}}\)之解\((x,y)\)為 。
[解答]
\(\displaystyle z=\frac{6}{4x+3y} \)
\(\displaystyle \frac{4}{x}=1-2z+z^2 \)
\(\displaystyle \frac{16}{y}=1+2z+z^2 \)
\(\displaystyle \frac{16}{y}-\frac{4}{x}=4z \)
\(\displaystyle \frac{4x-y}{xy}=\frac{6}{4x+3y} \)
\(\displaystyle 16x^2+8xy-3y^2=6xy \)
\(\displaystyle (8x-3y)(2x+y)=0 \)
\(\displaystyle 8x=3y \)
\(\displaystyle 1-\frac{6}{12x}=\frac{2}{\sqrt x} \)
\(\displaystyle 2x-4\sqrt x-1=0 \)
\(\displaystyle \sqrt x=\frac{2+\sqrt6}{2} \)
\(\displaystyle x=\frac{5+2\sqrt6}{2} \)
\(\displaystyle y=\frac{20+8\sqrt6}{3} \)
第8題
令\(\Delta ABC\)的外心為\(O\),垂心為\(H\)。若\(\Delta ABC\)的外接圓半徑為3,\(∠A=60^{\circ}\),\(∠B=45^{\circ}\),則\(\overline{OH}^2\)之值為 。
[解答]
由尤拉線性質知道
\(\displaystyle \vec{OH}=3\vec{OG}=\vec{OA}+\vec{OB}+\vec{OC} \)
然後就平方計算