The equation of a line in the general form can be written as ax +by+c = 0
1. Slope Intercept Form :
The equation of a line with slope m and making an intercept c on Y-axis is y = mx+c
2. General Form to Slope Intercept Form :
The equation of a line which passes through the point `(x _1, y_1)` and has the slope `'m'` is
`y-y_1=m(x-x_1)`
Let `Q(x_1, y_1)` be the point through which the line passes and let `P(x, y)` be any point on the line. Then, the
slope of the line is
`(y-y_1)/(x-x_1)`
But `m ` is the slope of the line. Therefore
`m=(y-y_1)/(x-x_1) => y -y_1=m(x-x_1)`
Thus, `y - y_1 = m(x - x_1)` is the required equation of the line.
3. Point Slope Form :
The equation of a line which passes through the point `(x_1, y_1 )` and has the slope m, is `(y- y_1) = m(x- x_1)`
4. Two Points Form :
The equation of a line passing through two points `(x_1, y_1)` and `(x_2, y_2 )` is `( y - y_1) = ((y_2 - y_1 )/( x_2 - x_1 ) ) ( x - x_1)`
or ` | (x ,y , 1), ( x_1 , y_1 , 1), (x_2, y_2 , 1)| = 0`
5. Intercept Form of a Line :
The equation of a line which cuts-off intercepts a and b respectively from X and Y -axes, is `x/a + y/b =1`
6. General Equation of a Line to Intercept Form :
The general equation of a line Ax+ By+ C = 0 is `x/(-(C/A) ) + y/(-(C/B)) =1`
7. Normal or Perpendicular Form :
The equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle a with X -axis is
`xcosalpha + ysinalpha = p`, where `0 < = alpha <= pi`
8. General Equation of a Line to the Normal Form :
The general equation of a line is Ax+ By+ C = 0.
Now, to reduce the general equation of a line to normal form, we first shift the constant term on the RHS and make it positive, if it is not so, then divide both sides by
`sqrt (text(coefficient of x)^2 + text( coefficient of y) ^2)`
`=> (A/sqrt(A^2 +B^2) ) x + (B/sqrt(A^2 + B^2 ) )y = ( (-C)/sqrt (A^2 + B^2 ) )`
9. Distance Point Form :
The equation of the straight line passing through `(x_1 ,y_1 )` and making an angle `theta` with the positive direction of X -axis is
`(x -x_1 ) /(cos theta ) = ( y - y_1 ) /(sin theta ) = r`
where, r is the distance of the point (x, y) on the line from the point `(x_1, y_1 )`.
The equation of a line in the general form can be written as ax +by+c = 0
1. Slope Intercept Form :
The equation of a line with slope m and making an intercept c on Y-axis is y = mx+c
2. General Form to Slope Intercept Form :
The equation of a line which passes through the point `(x _1, y_1)` and has the slope `'m'` is
`y-y_1=m(x-x_1)`
Let `Q(x_1, y_1)` be the point through which the line passes and let `P(x, y)` be any point on the line. Then, the
slope of the line is
`(y-y_1)/(x-x_1)`
But `m ` is the slope of the line. Therefore
`m=(y-y_1)/(x-x_1) => y -y_1=m(x-x_1)`
Thus, `y - y_1 = m(x - x_1)` is the required equation of the line.
3. Point Slope Form :
The equation of a line which passes through the point `(x_1, y_1 )` and has the slope m, is `(y- y_1) = m(x- x_1)`
4. Two Points Form :
The equation of a line passing through two points `(x_1, y_1)` and `(x_2, y_2 )` is `( y - y_1) = ((y_2 - y_1 )/( x_2 - x_1 ) ) ( x - x_1)`
or ` | (x ,y , 1), ( x_1 , y_1 , 1), (x_2, y_2 , 1)| = 0`
5. Intercept Form of a Line :
The equation of a line which cuts-off intercepts a and b respectively from X and Y -axes, is `x/a + y/b =1`
6. General Equation of a Line to Intercept Form :
The general equation of a line Ax+ By+ C = 0 is `x/(-(C/A) ) + y/(-(C/B)) =1`
7. Normal or Perpendicular Form :
The equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle a with X -axis is
`xcosalpha + ysinalpha = p`, where `0 < = alpha <= pi`
8. General Equation of a Line to the Normal Form :
The general equation of a line is Ax+ By+ C = 0.
Now, to reduce the general equation of a line to normal form, we first shift the constant term on the RHS and make it positive, if it is not so, then divide both sides by
`sqrt (text(coefficient of x)^2 + text( coefficient of y) ^2)`
`=> (A/sqrt(A^2 +B^2) ) x + (B/sqrt(A^2 + B^2 ) )y = ( (-C)/sqrt (A^2 + B^2 ) )`
9. Distance Point Form :
The equation of the straight line passing through `(x_1 ,y_1 )` and making an angle `theta` with the positive direction of X -axis is
`(x -x_1 ) /(cos theta ) = ( y - y_1 ) /(sin theta ) = r`
where, r is the distance of the point (x, y) on the line from the point `(x_1, y_1 )`.