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"The Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware 2
Basic combing
1. Determining whether two straight lines are parallel or perpendicular
(1) Suppose the slopes of the two straight lines l1 and l2 are k1 and k2 respectively, and the inclination angles are α1 and α2 respectively. Then when l1∥l2, α1=α2, thus l1∥l2⇔______. This is for non-overlapping In terms of straight lines l1 and l2. If it is not certain whether l1 and l2 overlap, when k1=k2, we can get ______ or __________.
(2) If both straight lines have slopes, and the slopes of l1 and l2 are k1 and k2 respectively, then l1⊥l2⇔__________. If the slope of l1 is 0, when l1⊥l2, the slope of l2 is ______, which The tilt angle is____.
Thoughts and insights
The necessary and sufficient condition for two straight lines l1 and l2 to be vertical is that the product of their slopes is -1. Is this correct?
Tip: Incorrect. The product of the slopes of two straight lines is -1, which means that the two straight lines are perpendicular. On the contrary, if the two straight lines are perpendicular, the product of the slopes is not necessarily -1. If the slope of one of the straight lines in l1 and l2 does not exist, the slope of the other straight line does not exist. When the slope is zero, l1 and l2 are perpendicular to each other.
2. The coordinates of the intersection of two straight lines
Two straight lines are known: l1: A1x＋B1y＋C1＝0, l2: A2x＋B2y＋C2＝0. When the condition_____________ is met, l1 and l2 intersect, and their intersection point can be obtained from the system of equations A1x＋B1y＋C1＝0A2x＋B2y＋C2＝0. If the system of equations has a solution, Then the two straight lines ______; if the system of equations has no solution, then the two straight lines ______; if the system of equations has countless solutions, then the two straight lines ______.
Warm up before class
1. (2010 College Entrance Examination Anhui Paper) The equation of the straight line passing through the point (1,0) and parallel to the straight line x-2y-2=0 is ()
A. x-2y-1=0B． x-2y+1＝0
C. 2x＋y－2＝0 D. x+2y-1＝0
2. "a=1" means "the straight line x+y=0 and the straight line x-ay=0 are perpendicular to each other" ()
A. sufficient but not necessary condition
B． necessary but not sufficient conditions
C. Necessary and Sufficient Condition
D. Neither sufficient nor necessary conditions
Breakthrough in test points
Parallel and perpendicularity of straight lines
1. For two non-overlapping straight lines l1 and l2, their slopes are k1 and k2 respectively, so l1∥l2⇔k1=k2.
2. If two straight lines have slopes and they are perpendicular to each other, then the product of their slopes is equal to -1; conversely, if the product of their slopes is equal to -1, then they are perpendicular to each other, that is, l1⊥l2⇔k1k2=-1.
3． Generally speaking, for two straight lines l1: A1x+B1y+C1=0, l2: A2x+B2y+C2=0, the judgment of the parallel relationship can be summarized as l1∥l2⇔A1B2-A2B1=0 and A1C2-A2C1≠0(B1C2-B2C1≠0); the vertical relationship can be It can be summarized as: l1⊥l2⇔A1A2+B1B2=0.
[Summary of rules] When using the slope-intercept formula of a straight line, y=kx+b, special attention should be paid to the special situation when the slope of the straight line does not exist. When using the general formula Ax+By+C=0 for a straight line, special attention should be paid to the special situation when A and B are zero.
When solving problems related to two straight lines being parallel or perpendicular, we mainly use the necessary and sufficient conditions for two straight lines to be parallel or perpendicular, that is, "equal slopes" or "negative reciprocals of each other." If the slope does not exist, you can consider using the method of combining numbers and shapes to study it.
Variation training 1: The straight line l passes through the point (-1,2) and is perpendicular to the straight line 2x-3y+4=0, then the equation of l is ()
A. 3x+2y-1=0B. 3x+2y+7＝0
C. 2x-3y+5=0 D. 2x-3y+8＝0
Example 1 (Bozhou survey in 2011) It is known that there are two straight lines l1: ax-by+4=0 and l2: (a-1)x+y+b=0. Find the values of a and b that satisfy the following conditions.
(1)l1⊥l2, and l1 passes the point (-3,-1);
(2)l1∥l2, and the distance from the coordinate origin to these two straight lines is equal.
[Idea Tip] Solve the system of equations based on the positional relationship between the two straight lines.
Example 2 (1) If the length of the line segment intercepted by the two parallel lines l1: x-y+1=0 and l2: x-y+3=0 is 2√2, then the inclination angle of m can be
①15° ②30° ③45° ④60° ⑤75°
The correct answer number is _________. (Write the serial numbers of all correct answers)
Methods and techniques
1. The positional relationship between two straight lines should consider parallelism, perpendicularity and coincidence. For two straight lines l1 and l2 whose slopes exist and do not overlap, l1∥l2⇔k1=k2; l1⊥l2⇔k1·k2=-1. If the slope of one straight line does not exist, then the slope of the other straight line is What must be paid special attention to. (like example 1)
2. Symmetry problems generally involve converting line-to-line symmetry into point-to-point symmetry. Use coordinate transfer method. (like example 3)
Predictions from famous teachers
1. It is known that two straight lines y=ax-2 and y=(a+2)x+1 are perpendicular to each other, then a is equal to ()
A. 2B. 1
C. 0D. -1
Analysis: Choose D. Method 1: Substitute the options into the question stem and observe.
It is easy to conclude that D meets the requirements.
Method 2: ∵ Lines y=ax-2 and y=(a+2)x+1 are perpendicular to each other,
∴a·(a+2)=-1. ∴a=-1. Therefore, choose D.
2. The distance from point (5,5) to straight line x+2y-5=0 is ()
A. 5B． 35
C. 25D.5
3． The number of straight lines that are equidistant from A(-1,-1) and B(2,2) and equal to 322 is ________.
Analysis: There are three in total, two of which are parallel to the straight line AB, and one passes through the midpoint of AB and is perpendicular to the straight line AB.
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"The Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware 4:
"Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware 4 Thoughts and Insights 1. The necessary and sufficient condition for the two straight lines l1 and l2 to be vertical is that the product of the slopes is -1. Is this correct? Tip: Incorrect. When the slopes of the two straight lines l1 and l2 do not exist, then the two...
"The Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware 3:
"The Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware 3 [Course Standard Requirements] 1. Can determine whether two straight lines are parallel or perpendicular based on their slope. 2. Can find the equation of a straight line based on whether two straight lines are parallel or perpendicular. 【Core scan】 1． Use two parallel lines...
"The Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware:
"The Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware Return to Textbook 1. If two straight lines have one and only one common point, then the two straight lines intersect; if there is no common point, then the two straight lines are parallel; if there are countless common points, then The two straight lines coincide. The straight line equation (1) passes...