(i) Circle having Centre `(h, k)` and Radius `r`
Equation of the circle is `(x -h)^2 + (y- k )^2 = r^2`
(ii) Circle having Centre `(0,0)` and Radius `r`
Equation of the circle is `x ^2 + y^2 = r ^2`
(iii) `text(Centre at X -axis)` y-coordinate of
centre `= 0`, i.e. `f = 0`
Centre `=(-g,0)`
Equation of the circle is
`x^2 + y^2 + 2gx + c = 0`
(iv) `text(Centre at Y-axis)` x-coordinate of centre `= 0`, i.e. `g = 0`
Centre `= (0,- f)`
Equation of the circle is
`x^2 + y^2 + 2fy + c = 0`
(v) `text(Centre at Origin)`
Centre `= (-g, -f)= (0, 0)`
Equation of the circle is `x^2 + y^2+ c = 0`
(vi) `text(Circle Passing through Origin)`
Equation of the circle is
`x ^2 + y^2 + 2gx + 2fy = 0`
because `(0, 0)` should satisfy the equation of the circle,
i.e. `c = 0`
(vii) `text(Circle Touching Y-axis at Origin and Centred at X-axis)`
Equation of the circle is
`x ^2 + y^2 -2gx =0`
(viii) `text(Circle Touching X -axis a Origin and Centred at Y-axis)`
Equation of the circle is `x^ 2 + y^2- 2fy = 0`
`(0, 0)` and `(0, -2 f)` are the end points of the diameter.
(ix) `text(Equation of a Circle in Diametric Form)` If `(x_1,y_1)` and `(x_2, y_2 )` are the end points of one of the diameter, then the equation of the circle is
`(x- x_1) (x- x_2) + (y- y_1) (y- y_2) = 0`
(x) `text(Parametric Form of a Circle)`
Consider the circle `(x - h )^2 + (y-k )^2 = r^2` centred at `A = (h, k)` and
radius `r`.
Let `P (x, y)`, the coordinates of P can be expressed as `x = h + r cos theta` and `y= h + r sin theta`
These equations represent the coordinates of any point on the circle in terms of the parameter `theta`.
(i) Circle having Centre `(h, k)` and Radius `r`
Equation of the circle is `(x -h)^2 + (y- k )^2 = r^2`
(ii) Circle having Centre `(0,0)` and Radius `r`
Equation of the circle is `x ^2 + y^2 = r ^2`
(iii) `text(Centre at X -axis)` y-coordinate of
centre `= 0`, i.e. `f = 0`
Centre `=(-g,0)`
Equation of the circle is
`x^2 + y^2 + 2gx + c = 0`
(iv) `text(Centre at Y-axis)` x-coordinate of centre `= 0`, i.e. `g = 0`
Centre `= (0,- f)`
Equation of the circle is
`x^2 + y^2 + 2fy + c = 0`
(v) `text(Centre at Origin)`
Centre `= (-g, -f)= (0, 0)`
Equation of the circle is `x^2 + y^2+ c = 0`
(vi) `text(Circle Passing through Origin)`
Equation of the circle is
`x ^2 + y^2 + 2gx + 2fy = 0`
because `(0, 0)` should satisfy the equation of the circle,
i.e. `c = 0`
(vii) `text(Circle Touching Y-axis at Origin and Centred at X-axis)`
Equation of the circle is
`x ^2 + y^2 -2gx =0`
(viii) `text(Circle Touching X -axis a Origin and Centred at Y-axis)`
Equation of the circle is `x^ 2 + y^2- 2fy = 0`
`(0, 0)` and `(0, -2 f)` are the end points of the diameter.
(ix) `text(Equation of a Circle in Diametric Form)` If `(x_1,y_1)` and `(x_2, y_2 )` are the end points of one of the diameter, then the equation of the circle is
`(x- x_1) (x- x_2) + (y- y_1) (y- y_2) = 0`
(x) `text(Parametric Form of a Circle)`
Consider the circle `(x - h )^2 + (y-k )^2 = r^2` centred at `A = (h, k)` and
radius `r`.
Let `P (x, y)`, the coordinates of P can be expressed as `x = h + r cos theta` and `y= h + r sin theta`
These equations represent the coordinates of any point on the circle in terms of the parameter `theta`.