A circle S passes through the point (0, 1) and is orthogonal to the circles `(x -1)^2 + y^2 = 16` and `x^2 + y^2 = 1`. Then (A) radius of S is 8 (B) radius of S is 7 (C) center of S is (-7,1) (D) center of S is (-8,1)

Options

A. radius of S is 8
B. radius of S is 7
C. centre of S is `( -7,1)`
D. centre of S is `( -8,1)`

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Identify the equations of the given circles
The two circles given in the problem are:
1. Circle 1: \((x - 1)^2 + y^2 = 16\)
2. Circle 2: \(x^2 + y^2 = 1\)

We can rewrite these equations in standard form:
- Circle 1: \(x^2 + y^2 - 2x - 15 = 0\) (Center: (1, 0), Radius: 4)
- Circle 2: \(x^2 + y^2 - 1 = 0\) (Center: (0, 0), Radius: 1)

### Step 2: Use the condition of orthogonality
For two circles to be orthogonal, the following condition must hold:
\[ S_1 - S_2 = 0 \]
Where \(S_1\) and \(S_2\) are the equations of the circles.

Substituting the equations:
\[ (x^2 + y^2 - 2x - 15) - (x^2 + y^2 - 1) = 0 \]
This simplifies to:
\[ -2x - 15 + 1 = 0 \implies -2x - 14 = 0 \implies x = -7 \]

### Step 3: Determine the center of circle S
We have found that the x-coordinate of the center of circle S is \(-7\). Let the center be \((-7, k)\).

### Step 4: Use the point through which circle S passes
Circle S passes through the point \((0, 1)\). We can use the distance formula to find the radius \(r\):
\[ r = \sqrt{(-7 - 0)^2 + (k - 1)^2} = \sqrt{49 + (k - 1)^2} \]

### Step 5: Express the radius in terms of the other circles
The radius can also be expressed using the coefficients of the standard form of the circles:
\[ r = \sqrt{g^2 + f^2 - c} \]
For circle 1 (the first circle):
- \(g = -2\), \(f = 0\), \(c = -15\)
Thus,
\[ r = \sqrt{(-2)^2 + 0^2 - (-15)} = \sqrt{4 + 15} = \sqrt{19} \]

### Step 6: Set the two expressions for radius equal
Since both expressions represent the same radius, we equate them:
\[ \sqrt{49 + (k - 1)^2} = \sqrt{19} \]
Squaring both sides gives:
\[ 49 + (k - 1)^2 = 19 \]
This simplifies to:
\[ (k - 1)^2 = 19 - 49 = -30 \]
This indicates a mistake in our calculations. Let's correct this by using the correct radius from the orthogonality condition.

### Step 7: Correctly calculate the radius
Using the distance from the center to the point (0, 1):
\[ r = \sqrt{(-7 - 0)^2 + (k - 1)^2} = \sqrt{49 + (k - 1)^2} \]
And from the orthogonality condition, we can also derive:
\[ r^2 = 49 + k^2 - 1 \]
Equating both expressions:
\[ 49 + (k - 1)^2 = 49 + k^2 - 1 \]
This simplifies to:
\[ (k - 1)^2 = k^2 - 1 \]
Expanding gives:
\[ k^2 - 2k + 1 = k^2 - 1 \]
Thus:
\[ -2k + 1 + 1 = 0 \implies -2k + 2 = 0 \implies k = 1 \]

### Step 8: Determine the radius
Now substituting \(k = 1\) back into the radius formula:
\[ r = \sqrt{49 + (1 - 1)^2} = \sqrt{49} = 7 \]

### Conclusion
The radius of circle S is \(7\), and the center is \((-7, 1)\).

### Final Answers
- (A) Radius of S is 8: **False**
- (B) Radius of S is 7: **True**
- (C) Center of S is (-7, 1): **True**
- (D) Center of S is (-8, 1): **False**