Question

Question
Which values of x and y satisfy the equation of the unit circle?(1 point)
Responses

x=2√2
, y=−2√2
x equals Start Fraction Start Root 2 End Root over 2 End Fraction , y equals negative Start Fraction Start Root 2 End Root over 2 End Fraction

x=0
, y=0
x equals 0 , y equals 0

x=−12
, y=32
x equals Start Fraction negative 1 over 2 End Fraction , y equals Start Fraction 3 over 2 End Fraction

x=3√2
, x=2√3

Answer 1

The equation of the unit circle is given by:

\[
x^2 + y^2 = 1
\]

To determine which pair of values \((x, y)\) satisfies this equation, we can check each response:

1. **Response 1:** \(x = 2\sqrt{2}, y = -2\sqrt{2}\)
\[
(2\sqrt{2})^2 + (-2\sqrt{2})^2 = 8 + 8 = 16 \quad \text{(not a solution)}
\]

2. **Response 2:** \(x = \frac{\sqrt{2}}{2}, y = -\frac{\sqrt{2}}{2}\)
\[
\left(\frac{\sqrt{2}}{2}\right)^2 + \left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1 \quad \text{(solution)}
\]

3. **Response 3:** \(x = 0, y = 0\)
\[
0^2 + 0^2 = 0 \quad \text{(not a solution)}
\]

4. **Response 4:** \(x = -\frac{1}{2}, y = \frac{3}{2}\)
\[
\left(-\frac{1}{2}\right)^2 + \left(\frac{3}{2}\right)^2 = \frac{1}{4} + \frac{9}{4} = \frac{10}{4} = 2.5 \quad \text{(not a solution)}
\]

5. **Response 5:** \(x = 3\sqrt{2}, y = 2\sqrt{3}\)
\[
(3\sqrt{2})^2 + (2\sqrt{3})^2 = 18 + 12 = 30 \quad \text{(not a solution)}
\]

Therefore, the only values that satisfy the equation of the unit circle are:

**\(x = \frac{\sqrt{2}}{2}, y = -\frac{\sqrt{2}}{2}\)**.