Definition of a Circle:
A circle is the locus of a point which moves in a plane such that its distance from a fixed point is always constant. The fixed point is called as centre of the circle and constant distance, radius of the circle.
The equation of a circle with the centre at (a, b) and radius r, is given by (x - a)2 + (y - b)2 = r2 .If centre of the circle is at the origin and radius is r, then equation of circle is x2 + y2 = r2.
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Condition
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Equation
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(i) Touches both the axes with centre (a, a) and radius a
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(x-a)2 + (y-a)2 = a2
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(ii) Touches x-axis only having centre (α, a) and radius a
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(x-α)2 + (y-a)2 = a2
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(iii) Touches y-axis only having centre (a, β) and radius a
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(x-α)2 + (y-β)2 = a2
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(iv) Passes through origin having centre
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x2 +ay2 - αx - βy = 0
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The general equation of 2nd degree ax2+2hxy+by2+2gx+2fy+c=0 represent a circle ,if
- Coefficient of x2=coefficient of y2 that is a=b
- Coefficient of xy=0 i.e. h=0.
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