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.

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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).

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.