The equation of a circle C is `x^(2)+y^(2)-6x-8y-11=0`. The number of real points at which the circle drawn with points (1, 8) and (0,0) as the ends of a diameter cuts the circle, C, is

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To solve the problem step by step, we will follow the outlined procedure:

### Step 1: Rewrite the equation of circle C
The given equation of circle C is:
\[ x^2 + y^2 - 6x - 8y - 11 = 0 \]

We can rearrange this equation into the standard form of a circle by completing the square.

### Step 2: Completing the square for x and y
1. For \(x\):
\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]
2. For \(y\):
\[ y^2 - 8y \rightarrow (y - 4)^2 - 16 \]

Substituting these back into the equation gives:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 - 11 = 0 \]
Simplifying this:
\[ (x - 3)^2 + (y - 4)^2 - 36 = 0 \]
Thus, we have:
\[ (x - 3)^2 + (y - 4)^2 = 36 \]
This means circle C has a center at (3, 4) and a radius of 6.

### Step 3: Find the equation of the circle with diameter endpoints (1, 8) and (0, 0)
The center of the circle formed by the endpoints (1, 8) and (0, 0) is the midpoint:
\[ \left(\frac{1 + 0}{2}, \frac{8 + 0}{2}\right) = \left(\frac{1}{2}, 4\right) \]
The radius is half the distance between the two points:
\[ \text{Distance} = \sqrt{(1 - 0)^2 + (8 - 0)^2} = \sqrt{1 + 64} = \sqrt{65} \]
Thus, the radius is \(\frac{\sqrt{65}}{2}\).

### Step 4: Write the equation of the second circle
Using the center and radius, the equation of the circle is:
\[ \left(x - \frac{1}{2}\right)^2 + (y - 4)^2 = \left(\frac{\sqrt{65}}{2}\right)^2 \]
This simplifies to:
\[ \left(x - \frac{1}{2}\right)^2 + (y - 4)^2 = \frac{65}{4} \]

### Step 5: Set up the equations to find intersection points
We now have two equations:
1. Circle C: \((x - 3)^2 + (y - 4)^2 = 36\)
2. Circle with diameter: \(\left(x - \frac{1}{2}\right)^2 + (y - 4)^2 = \frac{65}{4}\)

### Step 6: Subtract the two equations
Subtracting the equations helps eliminate \(y\):
\[ (x - 3)^2 - \left(x - \frac{1}{2}\right)^2 = 36 - \frac{65}{4} \]
Calculating the right side:
\[ 36 = \frac{144}{4} \implies \frac{144}{4} - \frac{65}{4} = \frac{79}{4} \]
Now, expanding the left side:
\[ (x^2 - 6x + 9) - \left(x^2 - x + \frac{1}{4}\right) = \frac{79}{4} \]
This simplifies to:
\[ -5x + \frac{35}{4} = \frac{79}{4} \]
Solving for \(x\):
\[ -5x = \frac{79}{4} - \frac{35}{4} = \frac{44}{4} = 11 \implies x = -\frac{11}{5} \]

### Step 7: Substitute \(x\) back to find \(y\)
Substituting \(x = -\frac{11}{5}\) into one of the circle equations (let's use Circle C):
\[ \left(-\frac{11}{5}\right)^2 + y^2 - 6\left(-\frac{11}{5}\right) - 8y - 11 = 0 \]
Calculating:
\[ \frac{121}{25} + y^2 + \frac{66}{5} - 8y - 11 = 0 \]
Converting all terms to have a common denominator of 25:
\[ \frac{121}{25} + \frac{330}{25} - \frac{275}{25} - 8y = 0 \]
This simplifies to:
\[ y^2 - 8y + \frac{176}{25} = 0 \]

### Step 8: Calculate the discriminant
The discriminant \(D\) is given by:
\[ D = b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot \frac{176}{25} \]
Calculating:
\[ D = 64 - \frac{704}{25} = \frac{1600 - 704}{25} = \frac{896}{25} \]
Since \(D > 0\), this indicates two distinct real points of intersection.

### Final Answer
Thus, the number of real points at which the two circles intersect is **2**.

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