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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(*x _{1}*,

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

Squaring both sides: (*x* - *h*)^{2} + (*y* - *k*)^{2} = *r*^{2}

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} = *r*^{2}

Or *x*^{2} + *y*^{2} = *r*^{2}

**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} = r^{2}

Here:

h = 3

k = 2

r = 3

Thus, the equation of the given circle is:

(x - 3)^{2} + (y - 2)^{2} = 3^{2}

This equation can be written as x^{2} + y^{2} - 6x - 4y + 4 = 0

Therefore, equations of the given circle are (x - 3)^{2} + (y - 2)^{2} = 3^{2}, x^{2} + y^{2} - 6x - 4y + 4 = 0

Further rearranging the equation, we get *x*^{2} + *y*^{2} + 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 *x*^{2} + *y*^{2} + 2*g**x* + 2*fy* + 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* = √*g*^{2} + *f*^{2} - *c.*

If g^{2} + f^{2} - c ≥ 0, x^{2} + y^{2} + 2gx + 2fy + c = 0 represents a circle with centre (-g, -f) and radius √g^{2} + f^{2} - c.

If *g*^{2} + *f*^{2} - *c =* 0, *x*^{2} + *y*^{2} + 2*g**x* + 2*fy* + c = 0 represents a point circle.

If *g*^{2} + *f*^{2} - *c <* 0, *x*^{2} + *y*^{2} + 2*g**x* + 2*fy* + c = 0 represents an imaginary circle with real centre.

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(*x _{1}*,

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

Squaring both sides: (*x* - *h*)^{2} + (*y* - *k*)^{2} = *r*^{2}

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} = *r*^{2}

Or *x*^{2} + *y*^{2} = *r*^{2}

**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} = r^{2}

Here:

h = 3

k = 2

r = 3

Thus, the equation of the given circle is:

(x - 3)^{2} + (y - 2)^{2} = 3^{2}

This equation can be written as x^{2} + y^{2} - 6x - 4y + 4 = 0

Therefore, equations of the given circle are (x - 3)^{2} + (y - 2)^{2} = 3^{2}, x^{2} + y^{2} - 6x - 4y + 4 = 0

Further rearranging the equation, we get *x*^{2} + *y*^{2} + 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 *x*^{2} + *y*^{2} + 2*g**x* + 2*fy* + 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* = √*g*^{2} + *f*^{2} - *c.*

If g^{2} + f^{2} - c ≥ 0, x^{2} + y^{2} + 2gx + 2fy + c = 0 represents a circle with centre (-g, -f) and radius √g^{2} + f^{2} - c.

If *g*^{2} + *f*^{2} - *c =* 0, *x*^{2} + *y*^{2} + 2*g**x* + 2*fy* + c = 0 represents a point circle.

If *g*^{2} + *f*^{2} - *c <* 0, *x*^{2} + *y*^{2} + 2*g**x* + 2*fy* + c = 0 represents an imaginary circle with real centre.