Equation of a Circle
1st Class Maths・2 minutes read
The equation of a circle x squared plus y squared equals r squared represents a circle with a center at 0 0 and radius r, which changes when shifting the center to a b with radius r to x minus a squared plus y minus b squared equals r squared. Equations of circles can be found by substituting values into x minus a squared plus y minus b squared equals r squared, helping draw circles and calculate equations with given points, including finding the tangent's equation at a point on the circle using gradients and y equals mx plus c.
Insights
- Shifting the center of a circle changes its equation: Moving the circle's center to coordinates (a, b) with radius r transforms the equation from x squared plus y squared equals r squared to (x - a) squared plus (y - b) squared equals r squared.
- Tangent line to a circle: To find the equation of a tangent line at a point on the circle, calculate the gradient of the radius first and then use the negative reciprocal as the gradient in the equation y = mx + c.
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Recent questions
How is the equation of a circle represented?
In the form x squared plus y squared equals r squared.
How does shifting the center of a circle affect its equation?
Shifting the center to a point (a,b) changes the equation to x minus a squared plus y minus b squared equals r squared.
How can equations of circles be determined?
By substituting values into x minus a squared plus y minus b squared equals r squared.
What is the significance of deriving circles from equations?
Helps in sketching circles with radius information from the equation.
How can the equation of a tangent to a circle at a point be found?
Determine the gradient of the radius and use the negative reciprocal for the tangent's gradient in y equals mx plus c.
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Summary
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Equations and Tangents of Circles
- Equation of a circle: x squared plus y squared equals r squared represents a circle with center at 0 0 and radius r.
- Moving the circle's center changes the equation: Shifting the center to a b with radius r alters the equation to x minus a squared plus y minus b squared equals r squared.
- Finding equations of circles: Given centers and radii, equations can be determined by substituting values into x minus a squared plus y minus b squared equals r squared.
- Drawing circles from equations: Centers and radii from equations help in sketching circles, with the radius derived from the equation.
- Calculating circle equations with given points: Substituting known points into x minus a squared plus y minus b squared equals r squared helps find the circle's equation.
- Tangent to a circle at a point: To find the tangent's equation at a point on the circle, determine the gradient of the radius, then use the negative reciprocal for the tangent's gradient in y equals mx plus c.
