l1:ax+by+c1=0 l2:ax+by+c2=0 The distance is: the absolute value of (c1-c2) divided by the square root (a square plus b square)
Distance formula: d=|C1-C2|/√(A^2+B^2) The origin of the formula: Let the two straight line equations be Ax+By+C1=0, Ax+By+C2=0. The distance between two parallel straight lines is the distance from any point on one straight line to another straight line.
If the point P(a,b) is on the straight line Ax+By+C1=0, then Aa+Bb+C1=0 is satisfied, that is, Ab +Bb=-C1. From the point to the straight line distance formula, the distance from P to the straight line Ax+By+C2=0 is d=|Aa+Bb+C2|/√(A^2+B^2)=|-C1+C2|/√(A ^2+B^2)=|C1-C2|/√(A^2+B^2) Extended data: Introduction to the formula of distance from point to line: 1. General formula: Let the equation of straight line L be Ax+By+C =0, the coordinates of the point P are (Xo, Yo), then the distance from the point P to the straight line L is: z1/n, there is s=|(x1-x0,y1-y0,z1-z0)×(l,m,n)|/√(l²+m²+n²) d=√((x1-x0)²+ (y1-y0)²+(z1-z0)²-s²) Second, the extended formula: Formula ①: Let the equation of the straight line l1 be; The equation of the straight line l2 is the distance between the two parallel lines: Formula ②: Let The equation of the straight line l1 is; the equation of the straight line l2 is the angle between the two straight lines,
Distance formula: d=|C1-C2|/√(A^2+B^2)
Official Origin:
Let the equations of the two straight lines be Ax+By+C1=0, Ax+By+C2=0. The distance between two parallel straight lines is the distance from any point on one straight line to another straight line. If the point P(a,b) is on the straight line Ax+By+C1=0, then Aa+Bb+C1=0 is satisfied, that is, Ab +Bb=-C1.
From the point to the straight line distance formula, the distance from P to the straight line Ax+By+C2=0 is d=|Aa+Bb+C2|/√(A^2+B^2)=|-C1+C2|/√(A ^2+B^2)=|C1-C2|/√(A^2+B^2)
Extended information:
Point-to-line distance formula introduction:
1. General formula:
Suppose the equation of the straight line L is Ax+By+C=0, and the coordinates of the point P are (Xo, Yo), then the distance from the point P to the straight line L is:
Considering the point (x0, y0, z0) and the space straight line x-x1/l=y-y1/m=z-z1/n, there are s=|(x1-x0, y1-y0, z1-z0)×(l ,m,n)|/√(l²+m²+n²)
d=√((x1-x0)²+(y1-y0)²+(z1-z0)²-s²)
Second, the extension formula:
Formula ①: Let the equation of straight line l1 be ;
The equation of line l2 is then the distance between 2 parallel lines:
Formula ②: Let the equation of straight line l1 be ; the equation of straight line l2 is
Then the included angle of the 2 lines,
The main answer to the question is as follows, the proof is very clumsy, sorry.
Assuming that the equations of two straight lines are
Ax+By+C1=0
Ax+By+C2=0
, the distance formula is |C1-C2|/√(A²+B²)
Derivation: The distance between two parallel straight lines is the distance between two parallel straight lines. The distance from one point to another straight line, set the point P(a, b) on the straight line Ax+By+C1=0, then satisfy Aa+Bb+C1=0, that is, Ab+Bb=-C1, the distance from the point to the straight line The formula, the distance from P to the straight line Ax+By+C2=0 is
d=|Aa+Bb+C2|/√(A^2+B^2)=|-C1+C2|/√(A^2+B^ 2)
=|C1-C2|/√(A^2+B^2)
The distance between parallel lines is equal everywhere and never intersect
Let the equations of the two straight lines be ax+by+c1=0 ax+by+c2=0 The distance between two parallel lines is the distance from any point on one straight line to another straight line, set the point p(a, b) on the straight line ax On +by+c1=0, it satisfies aa+bb+c1=0, that is, ab+bb=-c1, from the point to the straight line distance formula, the distance from p to the straight line ax+by+c2=0 is d=|aa+ bb+c2|/√(a^2+b^2)=|-c1+c2|/√(a^2+b^2) =|c1-c2|/√(a^2+b^2)
l1:ax+by+c1=0
l2:ax+by+c2=0
The distance is: the absolute value of (c1-c2) divided by the square root (a square plus b square) Ask whether the XY of the two equations must be the same ? This answer was adopted by netizens