2018 AMC 10A 第 10 题

Question

设实数

x

满足

\sqrt{49-x^2}-\sqrt{25-x^2}=3\text{。}

求下式的值：

\sqrt{49-x^2}+\sqrt{25-x^2}\text{？}

Suppose that real number

x

satisfies

\sqrt{49-x^2}-\sqrt{25-x^2}=3.

What is the value of

\sqrt{49-x^2}+\sqrt{25-x^2}?

8

\sqrt{33}+3

9

2\sqrt{10}+4

12

把两个根式组成的差与和看作一对共轭式。

Treat the two radicals as conjugate-like quantities

将已知的差乘以所求的和。

Multiply the given difference by the desired sum

题中给出的差与所求的和互为共轭式，将两式相乘即可消去根号。

乘积为

49 - x^2 - 25 + x^2 = 24\text{。}

所以两式的乘积是

24

，所求的和为

24 \div 3 = 8

。

因此正确答案是 A。

Note that the left hand side of the equation and the desired expression are conjugates. Multiplying them would remove the square roots.

Multiplying them yields

49 - x^2 - 25 + x^2 = 24.

This means that the product of the values of the expressions is equal to

24.

The desired value is therefore

24 \div 3 = 8.

Thus, A is the correct answer.

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