Number Theory - AMC 10A, 2018 - Problem 10
Try this beautiful problem from AMC 10A, 2018 based on Number theory.
Problem - Number Theory
Let's try this problem number 10 from AMC 10A, 2018 based on Number Theory.
Suppose that the 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
- $\sqrt{3} 3+8$
- 9
- $2 \sqrt{10}+4$
- 12
Key Concepts
Number Theory
Real number
Square root
Check the Answer
Answer: 8
AMC 10 A - 2018 - Problem No.10
Book Recommendation
Mathematics can be fun by Perelman
Try with Hints
As a first hint we can start from here :
In order to get rid of the square roots, we multiply by the conjugate. Its value is the solution. The $x^2$ terms cancel out.
$\left(\sqrt{49-x^2}+\sqrt{25-x^2}\right)\left(\sqrt{49-x^2}\right)-\left(\sqrt{25-x^2}\right)$
$=49-x^2-25+x^2=24$
Given that $\left.\sqrt{49-x^2}\right)-\left(\sqrt{25-x^2}\right)=3$
$\sqrt{49-x^2}+\sqrt{25-x^2}=\frac{24}{3} \quad=8$