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.

·       x² + y² + 2gx + 2fy + c = 0 (general form) is the equation of circle with centre at (- g, - f) and radius (g² + f² - c)^1/2

Note: Conditions for be an equation of second degree in x and y to represent an equation of circle:

       a) it should be a equation of second degree in x and y.

       b) the coefficients of x² and y² should be equal.

       c) there should be no term involving the product xy.

·        (x - x₁) (x - x₂) + (y - y₁) (y - y₂) = 0 (Diameter form) is the equation of circle with (x₁, y₁)  and (x₂, y₂) are the coordinates of ends of a diameter.

           Equation of tangent and normal to the circle

·       Equation of tangent to the circle x² + y² = a² at (x₁, y₁) is

xx₁ + yy₁ = a²

·       Equation of tangent to the circle x² + y² + 2gx + 2fy + c = 0 at

(x₁, y₁) is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

·       The line y = mx ± a(1 + m²)^1/2 are always tangent to the circle x² + y² = a².

·       The equation of normal at any point (x₁, y₁) to the circle x² + y²= a² is xy₁ = yx_1.

·       The normal at any point to a circle always passes through the centre of the circle.

·       The equation of normal to the circle x² + y² + 2gx + 2fy + c = 0 at (x₁, y₁) is   x₁y + g(y - y₁) = xy₁ + f(x - x₁).