Question

12．設(shè)計(jì)一個(gè)伸縮變換，把橢圓$\frac{{x}^{2}}{1}$+$\frac{{y}^{2}}{16}$=1變成單位圓．

Answer 1

分析 設(shè) $\left\{\begin{array}{l}{x′=λx}\\{y′=μy}\end{array}\right.$，得$\left\{\begin{array}{l}{x=\frac{x′}{λ}}\\{y=\frac{y′}{μ}}\end{array}\right.$，代入 $\frac{{x}^{2}}{1}$+$\frac{{y}^{2}}{16}$=1后求得λ，μ值得答案．

解答 解：設(shè) $\left\{\begin{array}{l}{x′=λx}\\{y′=μy}\end{array}\right.$，得$\left\{\begin{array}{l}{x=\frac{x′}{λ}}\\{y=\frac{y′}{μ}}\end{array}\right.$，代入 $\frac{{x}^{2}}{1}$+$\frac{{y}^{2}}{16}$=1得：$\frac{（x′）^{2}}{\frac{{λ}^{2}}{16}}+\frac{（y′）^{2}}{{μ}^{2}}=1$，
∵橢圓$\frac{{x}^{2}}{1}$+$\frac{{y}^{2}}{16}$=1變換成單位圓，
∴$\frac{{λ}^{2}}{16}$=μ²=1，即λ=4，μ=1．
則可解得伸縮變換φ：$\left\{\begin{array}{l}{x′=4x}\\{y′=y}\end{array}\right.$．

點(diǎn)評(píng) 本題考查了伸縮變換，關(guān)鍵是對(duì)變換公式的理解與運(yùn)用，是基礎(chǔ)題．