The two circles `x^(2)+y^(2)-25=0`, and `x^(2)+y^(2)-26y+25=0` are such that they

Options

A. touch
B. cut orthogonally
C. one is inside the other
D. none

Answer

To solve the problem of determining the relationship between the two circles given by the equations \( x^2 + y^2 - 25 = 0 \) and \( x^2 + y^2 - 26y + 25 = 0 \), we will analyze their properties step by step.

### Step 1: Identify the equations of the circles

The first circle is given by:
\[ x^2 + y^2 - 25 = 0 \]
This can be rewritten as:
\[ x^2 + y^2 = 25 \]
This represents a circle with center at \( (0, 0) \) and radius \( r_1 = 5 \) (since \( r = \sqrt{25} = 5 \)).

The second circle is given by:
\[ x^2 + y^2 - 26y + 25 = 0 \]
We can rearrange this equation:
\[ x^2 + (y^2 - 26y + 25) = 0 \]
Completing the square for the \( y \) terms:
\[ y^2 - 26y = (y - 13)^2 - 169 \]
Thus, we can rewrite the second circle as:
\[ x^2 + (y - 13)^2 - 169 + 25 = 0 \]
This simplifies to:
\[ x^2 + (y - 13)^2 = 144 \]
This represents a circle with center at \( (0, 13) \) and radius \( r_2 = 12 \) (since \( r = \sqrt{144} = 12 \)).

### Step 2: Find the distance between the centers of the circles

The centers of the circles are:
- Center of Circle 1: \( (0, 0) \)
- Center of Circle 2: \( (0, 13) \)

The distance \( d \) between the centers is given by:
\[ d = \sqrt{(0 - 0)^2 + (13 - 0)^2} = \sqrt{0 + 169} = 13 \]

### Step 3: Check the condition for the circles to touch each other

For two circles to touch each other externally, the distance between their centers must be equal to the sum of their radii:
\[ d = r_1 + r_2 \]
Calculating the sum of the radii:
\[ r_1 + r_2 = 5 + 12 = 17 \]

### Step 4: Compare the distance with the sum of the radii

Now, we compare the distance \( d \) with the sum of the radii:
\[ d = 13 \quad \text{and} \quad r_1 + r_2 = 17 \]

Since \( 13 \neq 17 \), the circles do not touch each other externally.

### Conclusion

The two circles do not touch each other.