The center of a circle is at the origin on a coordinate grid. A line with a positive slope intersects the circle at (0, 7). Which statement must be true?
A. The circle has a radius greater than 7.
B. The circle has a radius equal to 7.
C. The slope of the line is equal to 7.
D. The slope of the line is not equal to 7.
Answer (2)
The center of the circle is at the origin, and the circle passes through (0, 7).
Calculate the radius r using the equation of the circle x 2 + y 2 = r 2 , substituting the point (0, 7) to find r = 7 .
Analyze the line intersecting the circle at (0, 7) with a positive slope m , described by the equation y = m x + 7 .
Conclude that the circle's radius must be 7: T h ec i rc l e ha s a r a d i u se q u a lt o 7 .
Explanation
Analyze the problem and data The center of the circle is at the origin (0, 0). A line with a positive slope intersects the circle at the point (0, 7). We need to determine which statement must be true based on this information.
Find the radius of the circle Let the equation of the circle be x 2 + y 2 = r 2 , where r is the radius. Since the point (0, 7) lies on the circle, we have 0 2 + 7 2 = r 2 , which means r 2 = 49 , so r = 7 . Therefore, the circle has a radius equal to 7.
Analyze the line The equation of the line is y = m x + c , where m is the slope and c is the y-intercept. Since the line intersects the circle at (0, 7), the y-intercept is 7, so c = 7 . Thus, the equation of the line is y = m x + 7 . Since the line has a positive slope, 0"> m > 0 .
Determine the correct statement The question asks which statement must be true. We know the radius must be 7. The slope can be any positive number, so it's not necessarily equal to 7, and it's not necessarily not equal to 7. Therefore, the only statement that must be true is that the circle has a radius equal to 7.
Final Answer The circle has a radius equal to 7.
Examples
Understanding circles and lines is crucial in many fields. For example, in architecture, knowing how lines intersect circles helps in designing curved structures and ensuring structural integrity. Imagine designing a circular window and needing to place a support beam that intersects the window at a specific point. Using these principles, you can calculate the exact placement and angle of the beam to provide the necessary support without compromising the window's design.
The only statement that must be true is that the circle has a radius equal to 7, as determined from the intersection point (0, 7). The radius is calculated from the circle's equation using this point. Therefore, the correct answer is option B.
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