Solve each problem. Find all values of b such that the straight line 3x - y = b touches the circle x^2 + y^2 = 25 at only one point.

Verified step by step guidance

1. Recognize that the line 3x - y = b is tangent to the circle x^2 + y^2 = 25 if it touches the circle at exactly one point.
2. Recall that the condition for tangency between a line and a circle is that the perpendicular distance from the center of the circle to the line equals the radius of the circle.
3. The center of the circle x^2 + y^2 = 25 is (0, 0) and the radius is 5.
4. Use the formula for the distance from a point (x_1, y_1) to a line Ax + By + C = 0, which is |Ax_1 + By_1 + C| / sqrt(A^2 + B^2).
5. Substitute the center (0, 0) into the distance formula for the line 3x - y = b, and set the distance equal to the radius 5 to find the values of b.

Key Concepts

Circle Equation

The equation of a circle in standard form is given by (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. In this case, the circle is defined by x² + y² = 25, indicating a center at (0, 0) and a radius of 5. Understanding this helps in visualizing the geometric relationship between the line and the circle.

Line Equation

The equation of a line can be expressed in slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept. The given line equation, 3x - y = b, can be rearranged to find its slope and intercepts. This is crucial for determining how the line interacts with the circle.

Tangency Condition

For a line to touch a circle at exactly one point, the distance from the center of the circle to the line must equal the radius of the circle. This condition can be mathematically expressed using the formula for the distance from a point to a line. Solving for b under this condition will yield the values where the line is tangent to the circle.