Question

设实数 $x$ 满足求的值。 $\sqrt{49-x^2}-\sqrt{25-x^2}=3.$ $\sqrt{49-x^2}+\sqrt{25-x^2}?$

Suppose that real number $x$ satisfies $\sqrt{49-x^2}-\sqrt{25-x^2}=3.$ What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}?$

$8$

$\sqrt{33}+8$

$9$

$2\sqrt{10}+4$

$12$

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

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.