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Circle & Conics

  • A circle is the locus of a point which moves in such a way that its distance from a fixed point is constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.
  • The equation of the circle with radius r and centre (0, 0) is x2+y2=r2.

  • The equation of the circle with centre (a, b) and radius r is x-a2+y-b2=r2.

  • General equation of the circle is x2+y2+2gx+2fy+c=0, where -g, -f is the centre and r=g2+f2-c is the radius of the circle.

  • The equation of a circle with Ax1, y1 and Bx2, y2 as the extremities of a diameter is x- x1x-x2 +y- y1y-y2=0.

  • The equation of the circle with radius r, touching both the axes and lying in the first quadrant is x-r2+y-r2=r2.
  •  The equation of the circle with centre (a, b) and touching the x-axis only is x-a2+y-b2=b2.
  •  The equation of the circle with centre (a, b) and touching the y-axis only is x-a2+y-b2=a2.
  • The parametric equation of the circle x2+y2=r2 is x=rcosθ, y=rsinθ; 0θ2π. Any point on the circle x2+y2=r2 is given by rcosθ, rsinθ.
  • The parametric equation of the circle x-a2+y-b2=r2 is x=a+rcosθ, y=b+rsinθ; 0θ2π. Any point on the circle x-a2+y-b2=r2 is given by a+rcosθ, b+rsinθ.
 
Let Sx2+y2+2gx+2fy+c=0 be the general equation of a circle. Let …

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