After completion of this chapter, students will be able to:

- know about the angles between two lines, pair of straight lines and circle.
- solve the questions of angles between two lines.
- solve the questions of pair of straight lines.
- solve the theorems and questions of circle.

**The angle between two lines**

**Angle between the lines y = m _{1 }+ c_{1} and y = m_{2 + }c_{2} and y = m_{2}x + c_{2}**

**Condition of perpendicularity**

**m _{1}m_{2} = -1**

**Condition of Parallelism**

m_{1 }= m_{2}

Angle between the lines A_{1}x + B_{1}y + C_{1} = 0 and A_{2}x + B_{2}y +C_{2} = 0

**Condition of perpendicularity**

A_{1}A_{2} + B_{1}B_{2} = 0

**Condition of Parallelism**

A_{1}B_{2 }= A_{2}B_{1}

Equation of any line parallel to ax + by + c = 0

k = -bc

Equation of any line perpendicular to ax +by +c = 0

k = ac

General equation of the second degree :

ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0.

A circle is a closed curve such that every point on the curve is at a constant distance from a fixed point.

**Centre at the origin (Standard form)**

** x ^{2} + y^{2} = r^{2}**

**Centre at any point (Central form)**

**(x - h) ^{2} + (y k)^{2} = r^{2}**

**Circle with a given diameter (Diameter form) ^{}**

**(x - x _{1}) . (x - x_{2}) +(y - y_{1}) . (y - y_{2}) = 0**

**General equation of the circle**

**x ^{2} + y^{2} + 2gx + 2fy + c = 0**

**Equation of a circle in particular cases**

**When a circle touches the X - axis**

**(x - h) ^{2 }+ (y - k)^{2}= k^{2}**

**When a circle touches the Y- axis**

**(x - h) ^{2} + (y - k)^{2} = h^{2}**

**When a circle touches both the positive axes**

**(x - r) ^{2} + (y - k)^{2} = k^{2}**