For the circle `x^(2) + y^(2) = 25,` the point (-2.5, 3.5) lies :

Options

A. Inside circle
B. Outside circle
C. On the circle
D. none of these.

Answer

To determine the position of the point (-2.5, 3.5) relative to the circle defined by the equation \(x^2 + y^2 = 25\), we can follow these steps:

### Step 1: Identify the Circle's Equation
The equation of the circle is given as:
\[ x^2 + y^2 = 25 \]
Here, the radius \(r\) of the circle is \(\sqrt{25} = 5\).

### Step 2: Substitute the Point into the Circle's Equation
We need to check whether the point (-2.5, 3.5) lies inside, outside, or on the circle. We do this by substituting the coordinates of the point into the left-hand side of the circle's equation:
\[ s_1 = (-2.5)^2 + (3.5)^2 \]

### Step 3: Calculate the Values
Calculating the squares:
\[ (-2.5)^2 = 6.25 \]
\[ (3.5)^2 = 12.25 \]
Now, add these values together:
\[ s_1 = 6.25 + 12.25 = 18.5 \]

### Step 4: Compare with \(r^2\)
Now, we compare \(s_1\) with \(r^2\):
\[ r^2 = 25 \]
Since \(s_1 = 18.5\) and \(r^2 = 25\), we have:
\[ s_1 < r^2 \]

### Step 5: Determine the Position of the Point
According to the conditions:
- If \(s_1 < r^2\), the point lies inside the circle.
- If \(s_1 = r^2\), the point lies on the circle.
- If \(s_1 > r^2\), the point lies outside the circle.

Since \(s_1 = 18.5 < 25\), we conclude that the point (-2.5, 3.5) lies **inside** the circle.

### Final Answer
The point (-2.5, 3.5) lies **inside** the circle \(x^2 + y^2 = 25\).

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