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
= yx1.
· 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).
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