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4. (5 分) 已知椭圆 $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$ 的一个焦点为 $(2,0)$, 则 $C$ 的离心率为 ( )

Gaokao · Math · previous-year question

  1. A.$\frac{1}{3}$
  2. B.$\frac{1}{2}$
  3. C.$\frac{\sqrt{2}}{2}$correct
  4. D.$\frac{2 \sqrt{2}}{3}$

Answer

C. $\frac{\sqrt{2}}{2}$

Explanation

解：椭圆 $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$ 的一个焦点为 $(2,0)$, 可得 $a^{2}-4=4$, 解得 $a=2 \sqrt{2}$, $\because c=2$, $\therefore \mathrm{e}=\frac{\mathrm{c}}{\mathrm{a}}=\frac{2}{2 \sqrt{2}}=\frac{\sqrt{2}}{2}$. 故选: C.

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