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Circle

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Circle - Lesson Summary

When a plane intersects the nappe of a cone at any place other than the vertex, such that the angle between the plane and the axis of the cone is 90° the section of the cone is a circle.

A circle is the set of all the points in a plane that are equidistant from a fixed point. The fixed point is called the centre of the circle. The distance between any point on the circle and the centre is called the radius of the circle.

A circle can have an infinite number of radii and all its radii are equal.

Equation representing a circle on a coordinate plane:

Let a point P with the coordinates h and k be the centre of a circle.

Consider a point M(x, y) on the circle.

PM = radius (r)

Distance between points A(x1, y1) and B(x2, y2) = √(x2 - x1)2 + (y2 - y1)2

By distance formula: PM=√(x - h)2 + (y - k)2 = r

Squaring both sides: (x - h)2 + (y - k)2 = r2

This equation represents a circle in a coordinate plane. h and k in the equation represent the coordinates of the centre of the circle and r represents its radius.

Equation of a circle with centre at (0, 0):

In this case both, the coordinates of the centre of the circle will be zero.

Putting the values of h and k as zero in the equation of the circle,

(x - 0)2 + (y - 0)2 = r2

Or x2 + y2 = r2

General equation of the circle

Ex: Equation of the circle with the centre at the point (3, 2), and radius 3 units:

Sol:

Equation of a circle:

(x - h)2 + (y - k)2 = r2

Here:

h = 3

k = 2

r = 3

Thus, the equation of the given circle is:

(x - 3)2 + (y - 2)2 = 32

This equation can be written as x2 + y2 - 6x - 4y + 4 = 0

Therefore, equations of the given circle are (x - 3)2 + (y - 2)2 = 32, x2 + y2 - 6x - 4y + 4 = 0

Further rearranging the equation, we get x2 + y2 + 2(-3)x + 2(-2)y + 4 = 0.

This is another form of the equation of a circle, called the general equation of the circle.

General equation of a circle can be written as x2 + y2 + 2gx + 2fy + c = 0.

'-g' and '-f' represent the coordinates of the centre of the circle, and c represents the constant in the equation.

In the general equation of a circle, radius r = √g2 + f2 - c.

If g2 + f2 - c ≥ 0, x2 + y2 + 2gx + 2fy + c = 0 represents a circle with centre (-g, -f) and radius √g2 + f2 - c.

If g2 + f2 - c = 0, x2 + y2 + 2gx + 2fy + c = 0 represents a point circle.

If g2 + f2 - c < 0, x2 + y2 + 2gx + 2fy + c = 0 represents an imaginary circle with real centre.

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