Circle
Consider a fixed point. Then, a circle is a closed curve around the point such that every point on the curve is at a constant distance from that point. In terms of locus, a circle may be defined as the locus of a point which moves so that its distance from a fixed point is constant.
The fixed point is called the centre and the constant distance is called the radius of the circle.
Equation of a Circle
Centre at the origin (Standard form)
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Let
Centre at any point (Central form)
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Let
Equation of the circle touching the x-axis
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Let
Equation of the circle touching the y-axis
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Let
Equation of the circle touching both axes
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Let
General equation of the circle
Consider an equation
Comparing equation
- If
, the radius is real, hence the equation gives a real geometric locus. - If
, the radius is zero. In this case, the circle is called a point circle. - If
, the radius is imaginary. In this case, we say that the equation represents a circle with a real centre and an imaginary radius.
The general equation of second degree
Circle with a given diameter (Diameter form)
Let
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