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Questions and Answers
Find the equation of a circle whose diameter is the line segment joining A(2,-3) and B(-1,4)?
The equation of a circle with diameter joining points A and B can be found using the midpoint formula and the distance formula. The center of the circle is the midpoint of the line segment AB, and the radius is half the length of AB.
What is the midpoint formula?
The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by $((x1 + x2)/2, (y1 + y2)/2)$.
What is the distance formula?
The distance formula calculates the distance between two points (x1, y1) and (x2, y2) as $\sqrt{(x2 - x1)^2 + (y2 - y1)^2}$.
What is the equation of a circle whose diameter is the line segment joining A(2,-3) and B(-1,4)?
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What is the center of the circle?
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What is the radius of the circle?
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Study Notes
Circle Equation and Properties
- To find the equation of a circle, we need to know the center and radius of the circle.
- The midpoint formula is used to find the midpoint of a line segment, which is also the center of the circle.
- The midpoint formula is: M = ((x1 + x2)/2, (y1 + y2)/2), where M is the midpoint, and (x1, y1) and (x2, y2) are the endpoints of the line segment.
- The distance formula is used to find the radius of the circle, which is the distance between the center and any point on the circle.
- The distance formula is: √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the two points.
- The equation of a circle with center (a, b) and radius r is: (x - a)^2 + (y - b)^2 = r^2.
- To find the equation of the circle, first find the midpoint of the line segment joining A(2, -3) and B(-1, 4) using the midpoint formula.
- The midpoint is: M = ((2 + (-1))/2, (-3 + 4)/2) = M = (1/2, 1/2).
- The center of the circle is (1/2, 1/2).
- Use the distance formula to find the radius of the circle, which is the distance between the center and any point on the circle, say A(2, -3).
- The radius is: √(((2) - (1/2))^2 + ((-3) - (1/2))^2) = √(49/4) = 7/2.
- Now, plug in the values of the center and radius into the equation of the circle to get: (x - (1/2))^2 + (y - (1/2))^2 = (7/2)^2.
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Description
"Equation of Circle from Diameter" Quiz: Test your skills in finding the equation of a circle using the diameter defined by two given points. Explore the midpoint and distance formulas to enhance your understanding of circle geometry.
