Notes On Circle - CBSE Class 11 Maths
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.

#### 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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