§3 A straight line on a plane
1. Equations and graphs of a straight line in a plane
Equations and Graphics 
Description 

[ oblique cut ] _{} 

k is the slope . If the intersection angle between the line and the x axis is a , then 0 ￡ a < p ._{} b is the vertical intercept




[ intercept ] _{}


a , b are the intercepts on the x  axis and y axis, respectively . The line passes through the points A ( a , 0) and B (0, b ) . Intersection angle with the x axis _{} or _{} 

[ dot slant ]_{}

k is the slope The line passes through the point M ( x _{0} , y _{0} ) and intersects the x axis at an angle _{}


equation 
with graphics 
Description 

[ twopoint type ] _{} or_{}


The straight line passes through two points M _{1} ( x _{1} , y _{1} ) and M _{2} ( x _{2, }y _{2} ) , Intersection angle with the x axis _{}


[ General formula ] _{} ( A , B , C are constants, A and B are not zero at the same time )


slope _{} Longitudinal intercept _{} [ Note ] The general equation can be transformed into the above four forms as needed


[ parameter ] _{} or _{} (  ∞ < t < ∞ ) 

slope_{} The straight line passes through the point M ( x _{0} , y _{0} ) and the intersection angle with the x axis is a


[ Polar Coordinate Form ]


O is the pole, Ox is the polar axis, p is the distance from the pole to the line . a is the angle between the polar axis and the vertical line leading from the pole to the line ( positive counterclockwise ) , j is any point on the line Polar angle of M, r is the vector radius of point M


[ normal type ] _{}

p is the length of the normal line ( the length of the vertical line from the origin O to the line ) , b is the intersection angle between the normal line and the x axis, and p and b are called the position parameters of the line . [ Note ] The general formula of a straight line can be transformed into a normal formula _{} In the formula, it is called the normalization factor of the straight line. When C < 0 ( or C = 0 and B > 0) , take the positive sign; when C > 0 ( or C = 0 and B < 0) , take the negative sign_{} 

[ vector ] _{}

The line passes through the end point of the vector radius r _{0} and is parallel to the known vector a


equation 
with graphics 
Description 

[ plural ]
( a ) _{}
( b )_{} _{}


(a) The line passes through the point z _{0} and intersects the x axis at an angle a
( b ) The straight line passes through two points z _{1} , z _{2} ( t is a real parameter ) 

The relationship between points and lines on a plane
Equations and Graphics 
Calculation formula and description 
[ distance of dotted line ] normal _{} general _{}

d _{method} =_{} _{} where d is the distance from point M ( x _{0} , y _{0} ) to straight line L 
[ Included angle between two straight lines ] L _{1} A _{1} x + B _{1} y + C _{1} = 0 The slope is k _{1} L _{2} A _{2} x + B _{2} y + C _{2} = 0 The slope is k _{2}
_{}_{}is the angle between the two straight lines ( positive when counterclockwise from L _{1} to L _{2} ), and is the intersection of the two straight lines_{}

_{} _{} _{} _{};_{} In particular, when ( or ) , L _{1} // L _{2} ; _{}_{}_{}_{} At that time , L _{1} coincides with L _{2} ;_{}_{}_{} When A _{1 }A _{2} + B _{1 }B _{2} = 0 ( or 1 + k _{1 }k _{2} = 0) , L _{1} ⊥ L _{2}

Equations and Graphics 
Calculation formula and description 
[ Conditions for straight line bundle × three straight lines copoint ] L _{l} ( A _{1 }x + B _{1 }y + C _{1} ) + l ( A _{2 }x + B _{2 }y + C _{2} ) = 0, ( l is a parameter,  ￥ < l < ￥ )

For a certain value of l, L l _{represents} a straight line passing through the intersection point G of two straight lines ( L _{1} and L _{2} ) . When l takes all values, the whole of the straight lines that L _{l} represents passing through G is called a straight line bundle, G is called the vertex ( or center ) of the bundle of lines._{}_{}_{}_{} Let L _{3} be A _{3 }x + B _{3 }y + C _{3 } = 0 , then the condition for the three straight lines L _{1} , L _{2} , and L _{3} to have the same point is the determinant _{} If the equation of the two straight lines is given in the normal form, then  l  is the ratio of the distance between any point on the straight line L _{l} and the two given straight lines, and the lines corresponding to l = 1 and l =  1 are the given two straight lines Bisector of included angle
