Question

15．在同一坐标系中，将椭圆$\frac{{x}^{2}}{16}$+$\frac{{y}^{2}}{25}$=1变换成单位圆的伸缩变换是（　　）

• A． φ：$\left\{\begin{array}{l}{x′=5x}\\{{y}^{′}=4y}\end{array}\right.$
• B． φ：$\left\{\begin{array}{l}{{x}^{′}=4x}\\{{y}^{′}=5y}\end{array}\right.$
• C． φ：$\left\{\begin{array}{l}{{x}^{′}=\frac{1}{4}x}\\{{y}^{′}=\frac{1}{5}y}\end{array}\right.$
• D． φ：$\left\{\begin{array}{l}{{x}^{′}=\frac{1}{5}x}\\{{y}^{′}=\frac{1}{4}y}\end{array}\right.$

Answer 1

分析 设$\left\{\begin{array}{l}{x′=λx}\\{y′=μy}\end{array}\right.$，得$\left\{\begin{array}{l}{x=\frac{x′}{λ}}\\{y=\frac{y′}{μ}}\end{array}\right.$，代入$\frac{{x}^{2}}{16}$+$\frac{{y}^{2}}{25}$=1后求得λ，μ值得答案．

解答 解：设$\left\{\begin{array}{l}{x′=λx}\\{y′=μy}\end{array}\right.$，则$\left\{\begin{array}{l}{x=\frac{x′}{λ}}\\{y=\frac{y′}{μ}}\end{array}\right.$，代入$\frac{{x}^{2}}{16}$+$\frac{{y}^{2}}{25}$=1得：
$\frac{（x′）^{2}}{16{λ}^{2}}+\frac{（y′）^{2}}{25{μ}^{2}}=1$，
∵椭圆$\frac{{x}^{2}}{16}$+$\frac{{y}^{2}}{25}$=1变换成单位圆，
∴16λ²=25μ²=1，即$λ=\frac{1}{4}，μ=\frac{1}{5}$．
则φ：$\left\{\begin{array}{l}{x′=\frac{1}{4}x}\\{y′=\frac{1}{5}y}\end{array}\right.$．
故选：C．

点评 本题考查了伸缩变换，关键是对变换公式的理解与运用，是基础题．