Circle

 Circle

A circle is the locus of a point which moves so that its distance from a fixed point, called centre, is equal to a given distance. The given distance is called the radius of the circle.

Different forms of equation of circle:

·         x² + y² = a²(standard form) is the equation of circle with centre at (0, 0) and radius a.

·         (x - h)² + (y - k)² = a² (central form) is the equation of circle with centre at     (h, k) and radius a.

    Equations of Circle

1.        Equation of circle with centre at origin and radius a (Standard Equation)

          Let O(0, 0) be the centre and a be the radius of circle. Let P(x, y) be any point on the circle. Then,

          OP = a

or,       OP² = a²

or,       (x - 0)² + (y- 0)² = a²

or,       x² + y² = a² which is the equation of circle with centre at origin and radius a.

2.        Equation of circle with centre at any point (h, k) and radius a

          Let C(h, k) be the centre and a be the radius of the circle. Let P(x, y) be any point on the circle, then

          CP = r

or,       Cp² = a².

or,       (x - h)² + (y - k)² = a² which is the equation of circle with centre at (h, k) and radius a.

3.        General equation of circle

The equation of the form

          x² + y² + 2gx + 2fy + c = 0 can be written as

          x² + 2gx + y² + 2fy = - c

or,       x² + 2gx + g² + y² + 2fy + f² = g² + f² - c

Note:

(i)    The general equation of second degree ax² + by² + 2hxy + 2gx + 2fy + c = 0 in x and y represents circle if a = b and h = 0.

ii)      If g² + f² - c > 0, the radius is real and circle is real, if g² + f² - c = 0, then the radius is zero and circle is point circle and if g² + f² - c < 0, then circle is called circle with real centre and with imaginary radius.

4.        Equation of circle in diameter form

          Let A(x₁, y₁) and B(x₂, y₂) be the coordinates of end points of a diameter AB of A circle. Let P(x, y) be any point on the circle, then Ð APB = 90^o. 

A line and a circle

Let y = mx + c                                             …(i)

be a line and

          x² + y² = a²                                       …(ii)

be an equation of circle.

If the line intersects the circle, then the point of intersection can be obtained by solving these two equations.

Now, from (i) and (ii), w e have

          x² + (mx + c)² = a²

or,       x² + m²x² + 2m cx + c² - a² = 0              

or,       (1 + m²) x² + 2mcx + c² - a² = 0               ….(iii)

This equation is quadratic in x, hence in general x has two values. Corresponding to two values of x, we can have two values of y and the intersection points are obtained.

The discriminant of quadratic equation is

          4c²m² - 4(1 + m²) (c² - a²)

          = 4c²m² - 4c² + 4a² - 4m²c² + 4m²a²

          = 4{a²(1 + m²) - c²}

The following three cases may arise:

i)      If c² < a²(1 + m²), i.e. discriminant is positive. The roots of the quadratic equation (iii) are real and distinct. Hence the line (i) and circle (ii) intersect at two distinct point.

ii)   If a²(1 + m²) = c², i.e. discriminant is zero. Hence the line meets the circle at two coincident point, i.e. the line will be tangent to the circle.

iii)   If c² > a² (1 + m²), i.e. discriminant is negative. The roots of equation (iii) are imaginary and hence the line meets the circle at imaginary points.