Solve each problem. Find all values of b such that the straight line 3x - y = b touches the circle x² + y² = 25 at only one point.

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Rewrite the equation of the line in slope-intercept form to express y in terms of x and b: $3x - y = b$ becomes $y = 3x - b$.

Substitute $y = 3x - b$ into the circle equation $x^2 + y^2 = 25$ to get an equation in terms of $x$ and $b$: $x^2 + (3x - b)^2 = 25$.

Expand the squared term and simplify the equation to form a quadratic in $x$: $x^2 + 9x^2 - 6bx + b^2 = 25$, which simplifies to $10x^2 - 6bx + (b^2 - 25) = 0$.

For the line to be tangent to the circle, the quadratic equation must have exactly one solution, so set the discriminant equal to zero: $\Delta = (-6b)^2 - 4 \cdot 10 \cdot (b^2 - 25) = 0$.

Solve the discriminant equation for $b$ to find all values of $b$ where the line touches the circle at exactly one point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of a Circle

The equation x² + y² = 25 represents a circle centered at the origin with radius 5. Understanding this standard form helps identify the circle's size and position, which is essential when analyzing how lines interact with it.

Equation of a Line and Slope-Intercept Form

The line 3x - y = b can be rewritten as y = 3x - b, showing its slope (3) and y-intercept (-b). Recognizing this form allows for easier substitution into the circle's equation to find intersection points.

Condition for Tangency Between a Line and a Circle

A line touches a circle at exactly one point if the system formed by their equations has exactly one solution. This occurs when the discriminant of the resulting quadratic equation equals zero, indicating a single point of contact (tangency).

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527

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