Circles
1.0 Definition
2.0 Equation of circle in various forms
2.1 Central Form
2.2 Standard Form
2.3 Parametric Form
2.4 General form of circle
2.4.1 Application of general form of circle
2.5 Diametric form of circle
3.0 Intercepts made by a circle on coordinate axis
4.0 Position of a point with respect to a circle
5.0 Maximum and minimum distance of a point from a circle
6.0 Intersection of a line and a circle
7.0 Length of intercept cutoff from a line by a circle or length of chord of a circle
8.0 Equation of tangent to a circle
8.1 Slope form
8.2 Point form
8.3 If a point outside the circle is given through which tangent to a circle passes, then
8.4 Parametric form
9.0 Tangents from a point to the circle
10.0 Length of tangent from a point to a circle
11.0 Common Tangents
12.0 Equation of common tangents
13.0 Pair of tangents
14.0 Normal to a circle at a given point
15.0 Common chord of two circles
16.0 Equation of chord joining two points on circle
17.0 Equation of chord of circle whose midpoint is given
18.0 Chord of contact
19.0 Orthogonal Circles
20.0 Director Circle
21.0 Family of circles
2.5 Diametric form of circle
2.2 Standard Form
2.3 Parametric Form
2.4 General form of circle
2.4.1 Application of general form of circle
2.5 Diametric form of circle
8.2 Point form
8.3 If a point outside the circle is given through which tangent to a circle passes, then
8.4 Parametric form
The equation of circle on the line segment joining $A\left( {{x_1},{y_1}} \right)$ and $B({x_2},{y_2})$ as diameter is 
$$\left( {x - {x_1}} \right)\left( {x - {x_2}} \right) + \left( {y - {y_1}} \right)\left( {y - {y_2}} \right) = 0$$
$$\left( {x - {x_1}} \right)\left( {x - {x_2}} \right) + \left( {y - {y_1}} \right)\left( {y - {y_2}} \right) = 0$$Proof: Let $A({x_1},{y_1})$ and $B({x_2},{y_2})$ be the end points of a diameter and let $P(x,y)$ be any point on the circle. Slope of $AP = \frac{{y - {y_1}}}{{x - {x_1}}}$ and slope of $BP = \frac{{y - {y_2}}}{{x - {x_2}}}$.
Since $\angle APB = {90^ \circ }$ [Angle subtended by the diameter on circumference is ${90^ \circ }$]
Therefore, Slope of $AP \times $Slope of $BP=-1$ $[{m_1} \times {m_2} = - 1{\text{ for two perpendicular lines]}}$
$$\frac{{y - {y_1}}}{{x - {x_1}}} \times \frac{{y - {y_2}}}{{x - {x_2}}} = - 1$$ or, $$\left( {x - {x_1}} \right)\left( {x - {x_2}} \right) + \left( {y - {y_1}} \right)\left( {y - {y_2}} \right) = 0$$
It can also be written as $${x^2} + {y^2} - x\left( {{x_1} + {x_2}} \right) - y\left( {{y_1} + {y_2}} \right) + {x_1}{x_2} + {y_1}{y_2} = 0$$ or, $${x^2} + {y^2} - x({\text{sum of abscissae)}} - y({\text{sum of ordinates)}} + {\text{product of abscissae}} + {\text{product of ordinates}} = 0$$
Question 5. The sides of a square are $x = 2,x = 3,y = 1\ and\ y = 2$. Find the equation of circle drawn on diagonals of the square as its diameter.
Solution: From the equations of line we can calculate the intersection points as shown in figure $8$. Let us assume 
that $BD$ be the diagonal of the square and diameter of the circle with coordinates of $B(2,2)$ and $D(3,1)$. Using diametric form of circle, the equation of circle is $$\left( {x - 2} \right)\left( {x - 3} \right) + \left( {y - 2} \right)\left( {y - 1} \right) = 0$$$${x^2} - 5x + 6 + {y^2} - 3y + 2 = 0$$$${x^2} + {y^2} - 5x - 3y + 8 = 0$$
that $BD$ be the diagonal of the square and diameter of the circle with coordinates of $B(2,2)$ and $D(3,1)$. Using diametric form of circle, the equation of circle is $$\left( {x - 2} \right)\left( {x - 3} \right) + \left( {y - 2} \right)\left( {y - 1} \right) = 0$$$${x^2} - 5x + 6 + {y^2} - 3y + 2 = 0$$$${x^2} + {y^2} - 5x - 3y + 8 = 0$$ Question 6. Point $A$ has coordinates $({x_1},{y_1})$ and point $B$ has coordinates $({x_2},{y_2})$. The quadratic equation ${x^2} + 2ax - {b^2} = 0$ has roots ${x_1}$ and ${x_2}$ and the quadratic equation ${x^2} + 2px - {q^2} = 0$ has roots ${y_1}$ and ${y_2}$. Find the equation of circle with $AB$ as diameter.
Solution: As ${x_1}$ and ${x_2}$ are the roots of quadratic equation ${x^2} + 2ax - {b^2} = 0$,
$${x_1} + {x_2} = - 2a...(1)$$ and $${x_1} \times {x_2} = - {b^2}...(2)$$
As ${x_1}$ and ${x_2}$ are the roots of quadratic equation ${x^2} + 2px - {q^2} = 0$,
$${y_1} + {y_2} = - 2p...(3)$$ and $${y_1} \times {y_2} = - {q^2}...(4)$$ Using diametric form, the equation of circle with $AB$ as diameter can be written as $$\left( {x - {x_1}} \right)\left( {x - {x_2}} \right) + \left( {y - {y_1}} \right)\left( {y - {y_2}} \right) = 0$$ $${x^2} - x{x_2} - {x_1}x + {x_1}{x_2} + {y^2} - y{y_2} - {y_1}y + {y_1}{y_2} = 0$$ $${x^2} - x\left( {{x_2} + {x_1}} \right) + {x_1}{x_2} + {y^2} - y\left( {{y_2} + {y_1}} \right) + {y_1}{y_2} = 0$$ Put the values from equations $(1),\ (2),\ (3)\ and\ (4)$, $${x^2} + {y^2} + 2ax + 2py - {b^2} - {q^2} = 0$$
