At how many points does the line y=2x-1 intersect the circle `(x-4)^(2)+(y-6)^(2)=2` ?

Answer

To determine how many points the line \( y = 2x - 1 \) intersects the circle given by the equation \( (x - 4)^2 + (y - 6)^2 = 2 \), we can follow these steps:

### Step 1: Substitute the line equation into the circle equation
We start by substituting \( y = 2x - 1 \) into the circle's equation.

The equation of the circle is:
\[ (x - 4)^2 + (y - 6)^2 = 2 \]

Substituting \( y \):
\[ (x - 4)^2 + ((2x - 1) - 6)^2 = 2 \]

### Step 2: Simplify the equation
Now we simplify the equation:
\[ (x - 4)^2 + (2x - 7)^2 = 2 \]

Expanding both squares:
\[ (x - 4)^2 = x^2 - 8x + 16 \]
\[ (2x - 7)^2 = 4x^2 - 28x + 49 \]

Combining these:
\[ x^2 - 8x + 16 + 4x^2 - 28x + 49 = 2 \]

### Step 3: Combine like terms
Now, combine the terms:
\[ 5x^2 - 36x + 65 = 2 \]

Subtract 2 from both sides:
\[ 5x^2 - 36x + 63 = 0 \]

### Step 4: Analyze the quadratic equation
We have a quadratic equation:
\[ 5x^2 - 36x + 63 = 0 \]

To determine the number of intersection points, we can calculate the discriminant \( D \) of the quadratic equation:
\[ D = b^2 - 4ac \]
where \( a = 5 \), \( b = -36 \), and \( c = 63 \).

Calculating the discriminant:
\[ D = (-36)^2 - 4 \cdot 5 \cdot 63 \]
\[ D = 1296 - 1260 \]
\[ D = 36 \]

### Step 5: Determine the number of intersections
Since the discriminant \( D > 0 \), this indicates that the quadratic equation has two distinct real roots. Therefore, the line intersects the circle at two points.

### Conclusion
The line \( y = 2x - 1 \) intersects the circle \( (x - 4)^2 + (y - 6)^2 = 2 \) at **two points**.

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