The point (2, 4) lies inside the circle `x^(2) + y^(2) = 16`. The above statement is

Options

A. Always false
B. Always true
C. Can be true
D. Cannot be determined

Answer

To determine whether the point (2, 4) lies inside, outside, or on the circle defined by the equation \(x^2 + y^2 = 16\), we can follow these steps:

### Step 1: Identify the Circle's Equation
The equation of the circle is given as:
\[ x^2 + y^2 = 16 \]
This represents a circle centered at the origin (0, 0) with a radius of 4 (since \( \sqrt{16} = 4\)).

### Step 2: Substitute the Point into the Circle's Equation
We need to check the position of the point (2, 4) relative to the circle. We will substitute \(x = 2\) and \(y = 4\) into the left side of the circle's equation:
\[ x^2 + y^2 = 2^2 + 4^2 \]

### Step 3: Calculate the Values
Now, we calculate:
\[ 2^2 = 4 \quad \text{and} \quad 4^2 = 16 \]
Adding these together gives:
\[ 4 + 16 = 20 \]

### Step 4: Compare with the Circle's Radius
Now we compare this result with the right side of the circle's equation:
\[ 20 \quad \text{(calculated value)} \quad \text{and} \quad 16 \quad \text{(circle's radius squared)} \]
Since \(20 > 16\), we conclude that:
\[ x^2 + y^2 > 16 \]

### Step 5: Determine the Position of the Point
According to the conditions for the position of a point relative to a circle:
- If \(x^2 + y^2 < 16\), the point lies inside the circle.
- If \(x^2 + y^2 = 16\), the point lies on the circle.
- If \(x^2 + y^2 > 16\), the point lies outside the circle.

Since we found that \(20 > 16\), we conclude that the point (2, 4) lies **outside** the circle.

### Conclusion
The statement that the point (2, 4) lies inside the circle \(x^2 + y^2 = 16\) is **false**.

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