1.If a line L makes an angle  \(\theta\)   \((0^0\le\theta <180^0)\)  with the positive direction  of the \(X-axis\)  then  \(\theta\)  is called inclination of L and   \(\tan \theta\)  is called the slope of the line L

The slope is generally denoted by m i.e  \(m=\tan \theta\)

2.   i) The slope of a horizontal line is  zero

     ii) The slope of the vertical line is  not defined

3. i) The equation of  \(x-axis\)   is  \(y=0\)  

    ii) The equation  of the y-axis is \(x=0\)

   iii) Equation of line parallel to the x-axis  is \(y=k\)

  iv) Equation of line parallel to y-axis is  \(x=k\)

1 a) If  \(A(x_1,y_1)\)   and   \(B(x_2,y_2)\)   are two points. then the slope of  \(\overline{AB}= \frac {y_2-y_1} {x_2--x_1}\)

    b) The slope of the line  \(ax+by+c=0\)  is \(-\frac a b\)

2  a) If  \(m_1,m_2\)  are slopes of two parallel lines then \(m_1=m_2\)

   b) If   \(m_1,m_2\)  are slopes of two perpendicular lines then \(m_1m_2=-1\Rightarrow m_2=-\frac {1} {m_1}\)

3  Slope point form:- The  equation of a line passing through  \((x_1,y_1)\)   and having  slope  \('m'\)   is   \(y-y_1=m(x-x_1)\)

 ii) The equation of a line passing through the origin and having slope 'm' is \(y=mx\)

4 Two points form:- The equation of the line passing through the points \((x_1,y_1)\) and  \((x_2,y_2)\)  is  \(y-y_1=\frac {y_2-y_1} {x_2-x_1} (x-x_1)\)

1 Slope intercept form:-

i) The equation  of the line with slope 'm' and y-intercept 'c' is \(y=mx+c\)

ii) The equation of the line with slope 'm' and x-intercept 'a' is \(y= m(x-a)\)

2 Intercept form 

The  equation of a line having 'a','b' as intercepts are \(\frac {x} {a}+\frac{y} {b}=1\)

3  Normal form 

  The equation  of a line in the normal form is  \(x \cos \alpha+y\sin \alpha =p\)

4 symmetric form 

The  equation of a line passing through \((x_1,y_1)\) and having inclination   \(\theta\) is

\(\frac {x-x_1} {\cos \theta}=\frac {y-y_1} {\sin \theta}\) 

1   Parametric form:-

The parametric equations of the line are   \(x=x_1\pm r \cos \theta\)  & \(y=y_1\pm r \sin \theta\)

2  (a) The perpendicular distance from the origin to the line \(ax+by+c=0\)  is \(\frac {|c|} {\sqrt{a^2+b^2}}\)

(b)The perpendicular distance from \(P(x_1,y_1)\) to the line \(ax+by+c=0\)  is

\(\frac {|ax_1+by_1+c|} {\sqrt{a^2+b^2}}\)

3 The distance  between the parallel lines  \(ax+by+c_1=0\) and  \(ax+by+c_2=0\)  is \(\frac {|c_1-c_2|}{\sqrt{a^2+b^2}}\)

4) (a) Area of the triangle formed by the line \(\frac {x} {a}+\frac {y} {b}=1\)  with co-ordinate axes are  \(\frac 1 2|ab|\)

(b) The area of the triangle  formed by the line \(ax+by+c=0\) with co-ordinate axes are  \(\frac {c^2} {2|ab|}\)

1  a) Area of the triangle formed by the line  \(\frac {x} {a}+\frac {y} {b}=1 \)  with coordinate axes is \(\frac 1 2 |ab|\)

    b) The area of a triangle formed by the line ax+by+c with co-ordinate axes is \(\frac {c^2} {2|ab|}\)

2 a) The x-axis  divides \(\overline{AB}\)   in the ratio \(-y_1:y_2\)

     b) The y-axis  divides \(\overline{AB}\)   in the ratio \(-x_1:x_2\)

3   i ) The ratio in which the line \(L\equiv ax+by+c=0\)  divides the line segment  joining \(A(x_1,y_1),B(x_2,y_2)\)  is \(-L_{11}:L_{22}\)  where  \(L_{11}\equiv ax_1+by_1+c=0\),\(L_{22}\equiv ax_2+by_2+c=0\)

ii) The  points  A,B lie on the same side or opposite side of the line \(L=0\) according  \(L_{11},L_{22}\)  have the same sign or opposite signs

1  If  \(m_1,m_2\)   are slopes of two lines and the angle between two lines is \(\theta\) then 

\(\tan \theta = \Big| \frac {m_1-m_2} {1+m_1m_2}\Big|\)

2  If  \(\theta\)  is acute angle  between the lines  \(a_1x+b_1y+c_1=0\)  and \(a_2x+b_2y+c_2=0\)  then 

\(\tan \theta=\Big|\frac {a_1b_2-a_2b_1} {a_1a_2+b_1b_2}\Big|\)

3 a) The equation of the line parallel to  \(ax+by+c=0\)  is  \(ax+by+k=0\)

   b) The  equation of the line passing through  \((x_1,y_1)\)  and parallel to \(ax+by+c=0 \) is \(a(x-x_1) +b(y-y_1)=0\)

    c) The equation of the line perpendicular to \(ax+by+c=0\)  is \(b(x-x_1) -a(y-y_1)=0\)