💻 Equation of a circle

This lesson covers core applied mathematics concepts and their agricultural applications for BSc Agriculture learners.

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MATHS :: Lecture 08 ::Equation of a circle

Circles

A circle is defined as the locus of the point, which moves in such a way, that its distance from a fixed point is always constant. The fixed point is called centre of the circle and the constant distance is called the radius of the circle.

The equation of the circle when the centre and radius are given

Let C (h,k) be the centre and r be the radius of the circle. Let P(x,y) be any point on the circle. CP = r CP2 = r2 (x-h)2 + (y-k)2 = r2 is the required equation of the circle.

Note : If the center of the circle is at the origin i.e., C(h,k)=(0,0) then the equation of the circle is x2 + y2 = 2 The general equation of the circle is x2 + y2 +2gx + 2fy + c = 0 Consider the equation x2 + y2 +2gx + 2fy + c = 0. This can be written as x2 + y2 + 2gx +2fy + g2 + f2 = g2 +f2 – c (i.e) x2 + 2gx + g2 + y2 +2fy + f2 = g2 +f2 – c (x + g)2 + (y + f )2 = + = This is of the form (x-h)2+ (y-k)2 = r2 \The considered equation represents a circle with centre (-g,-f) and radius \ The general equation of the circle is x2 + y2 +2gx + 2fy + c = 0 where c = The Center of the circle whose coordinates are (-g,-f) r = The radius of the circle =

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Note

The general second degree equation ax2 + by2 +2hxy + 2gx + 2fy +c = 0 Represents a circle if (i) a = b i.e., coefficient of x2 = coefficient of y2 (ii) h = 0 i.e., no xy term

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Summary Cheat Sheet

• Focus on core formulas, definitions, and solved patterns from this lesson.
• Practice stepwise derivations and numerical substitutions carefully.
• Connect each concept to practical agricultural problem-solving contexts.

References

1 source

Primary classroom notes and standard BSc Agriculture applied mathematics references.