Tht line L passes through the points f intersection of the circles `x ^(2) + y^(2) =25` and `x ^(2) + y^(2) -8x+7=0.` The length of perpendicular from center of second circle onto the line L is

Options

A. 4
B. 3
C. 1
D. 0

Answer

To solve the problem of finding the length of the perpendicular from the center of the second circle to the line L that passes through the points of intersection of the two circles, we will follow these steps:

### Step 1: Identify the equations of the circles
The first circle is given by:
\[ C_1: x^2 + y^2 = 25 \]
The second circle is given by:
\[ C_2: x^2 + y^2 - 8x + 7 = 0 \]

### Step 2: Rewrite the second circle in standard form
To find the center and radius of the second circle, we can rewrite its equation:
\[ x^2 + y^2 - 8x + 7 = 0 \]
Completing the square for the \(x\) terms:
\[ (x^2 - 8x + 16) + y^2 = 16 - 7 \]
This simplifies to:
\[ (x - 4)^2 + y^2 = 9 \]
Thus, the center of the second circle \(C_2\) is at \((4, 0)\) and the radius is \(3\).

### Step 3: Find the points of intersection of the circles
To find the intersection points, we can substitute \(y^2\) from the first circle into the second circle's equation:
From \(C_1\):
\[ y^2 = 25 - x^2 \]
Substituting into \(C_2\):
\[ x^2 + (25 - x^2) - 8x + 7 = 0 \]
This simplifies to:
\[ 25 - 8x + 7 = 0 \]
\[ -8x + 32 = 0 \]
\[ x = 4 \]

Now, substituting \(x = 4\) back into \(C_1\) to find \(y\):
\[ 4^2 + y^2 = 25 \]
\[ 16 + y^2 = 25 \]
\[ y^2 = 9 \]
Thus, \(y = 3\) or \(y = -3\).

The points of intersection are:
\[ (4, 3) \text{ and } (4, -3) \]

### Step 4: Find the equation of the line L
The line L passes through the points \((4, 3)\) and \((4, -3)\). Since both points have the same \(x\)-coordinate, the line is vertical:
\[ x = 4 \]

### Step 5: Calculate the perpendicular distance from the center of the second circle to line L
The center of the second circle is at \((4, 0)\). The line L is given by \(x = 4\). Since the center of the circle lies on the line \(x = 4\), the perpendicular distance from the center to the line is:
\[ \text{Distance} = 0 \]

### Final Answer
The length of the perpendicular from the center of the second circle onto the line L is:
\[ \boxed{0} \]