the degenerated case of a parabola. (c) When ≤ β < α , the section is a pair of intersecting straight lines (Fig11. ). It is the degenerated case of a hyperbola .
Fig . Fig . Fig . CONIC SECTIONS In the following sections, we shall obtain the equations of each of these conic sections in standard form by defining them based on geometric properties.
Circle Definition A circle is the set of all points in a plane that are equidistant from a fixed point in the plane. The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle (Fig . ). MATHEMATICS The equation of the circle is simplest if the centre of the circle is at the origin.
However, we derive below the equation of the circle with a given centre and radius (Fig . ). Given C ( h , k ) be the centre and r the radius of circle. Let P( x , y ) be any point on the circle (Fig11.
). Then, by the definition, | CP | = r . By the distance formula, we have x – h y – k i.e. ( x – h ) + ( y – k ) = r This is the required equation of the circle with centre at ( h , k ) and radius r .
Example Find an equation of the circle with centre at ( , ) and radius r . Solution Here h = k = . Therefore, the equation of the circle is x + y = r . Example Find the equation of the circle with centre (– , ) and radius .
Solution Here h = – , k = and r = . Therefore, the equation of the required circle is ( x + ) + ( y – ) = Example Find the centre