"The Positional Relationship of Two Straight Lines" Parallel Lines and Intersecting Lines PPT Courseware 3

Description

"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 the condition that two straight lines are parallel or perpendicular to solve the problem. (emphasis)

2. Often combined with the solution of straight line equations. (difficulty)

3． When the slope of the straight line does not exist, the positional relationship between the two straight lines is determined using the method of classification and discussion. (method)

Self-study guide

1. Two straight lines are parallel

(1) Two non-overlapping straight lines l1: y=k1x+b1 and l2: y=k2x+b2 (b1≠b2), if l1∥l2, then ____; conversely, if k1=k2, then ____, as shown in the figure.

(2) If the slopes of the two non-overlapping straight lines l1 and l2 do not exist, then their inclination angles are both ____, so they are parallel to each other.

Think about it: why are two straight lines with equal slopes not necessarily parallel?

Tip: We know that the geometric elements that determine the position of a straight line in a plane Cartesian coordinate system are: a fixed point on the straight line and its inclination angle. The slopes are equal, which means that their inclination angles are equal. However, straight lines with equal inclination angles are not necessarily parallel and may overlap. This is because it is necessary to determine whether they pass through a different fixed point. Usually it is enough to verify whether the intersection points of these two straight lines and the y-axis, that is, the intercepts on the y-axis, are equal.

2. Two straight lines are perpendicular

(1) Suppose straight line l1: y＝k1x＋b1, straight line l2: y＝k2x＋b2. If l1⊥l2, then k1·k2＝; conversely, if k1·k2＝-1, then .

(2) For straight line l1: x=a, straight line l2: y=b, since l1⊥x axis, l2⊥y axis, so l1⊥l2.

Question Type 1: Determining whether two straight lines are parallel or perpendicular

[Example 1] Based on the following conditions, determine the positional relationship between straight line l1 and straight line l2.

(1) l1: y=-3x+1.l2: x+1/3y-6=0;

(2) l1 passes through points A(-1,-2), B(2,1), l2 passes through points M(3,4), N(-1,-1);

(3) The slope of l1 is -10, and l2 passes through points A(10,2) and B(20,3);

(4) The inclination angle of l1 is 60°, the intercept on the y-axis is -2, the intercept of l2 on the x-axis is 3, and the intercept on the y-axis is √3.

[Idea Exploration] To answer this question, you can first find the equation of the straight line, then determine the slope of the straight line and the intercept on the y-axis, and use these two elements to determine the positional relationship between the two straight lines.

Solution (1) The slopes of the two straight lines are k1=-3, k2=-3, and the intercepts on the y-axis are b1=1, b2=18 respectively. Because k1=k2, b1≠b2, so l1∥ l2.

(2)k1＝1-(-2)/2-(-1)＝1, k2＝-1-4/-1-3＝54,

k1≠k2, ∴l1 and l2 are not parallel.

Question type 2: Use two straight lines to be parallel and perpendicular to find the equation of a straight line

[Example 2] (1) Find the equation of the straight line l parallel to the straight line 3x+4y+1=0 and passing through the point (1,2);

(2) Find the equation of the straight line l passing through the point A(2,1) and perpendicular to the straight line 2x+y-10=0.

Exploring ideas

To find a straight line that passes through a certain point and is parallel (or perpendicular) to the known straight line Ax+By+C=0 using the regular method, you can first find the slope of the straight line, and then use the point-slope formula to write the equation you are looking for. You can also set the equation you are looking for as Ax+By+m=0 (or Bx-Ay+m=0), and then use the undetermined coefficient method to solve it.

Question Type 3: Use the parallel or perpendicular relationship of straight lines to find parameters

[Example 3] (12 points) It is known that straight line l1 passes through points A(3, a), B(a-1,2), and straight line l2 passes through points C(1,2), D(-2, a+2).

(1) If l1∥l2, find the value of a;

(2) If l1⊥l2, find the value of a.

Question review guidance: Since the question contains parameters and involves the slope of a straight line, attention should be paid to classifying and discussing the presence or absence of the slope when solving the problem.

Methods and skills: Application of classification discussion ideas in linear position relationships

When the objects given in the question cannot be studied uniformly, the objects of study must be classified, and then each category is studied separately to obtain the results of each category. Finally, the results of each category are combined to obtain the answer to the entire problem. This is classification discussion. ． In this section, we mainly use the idea of ​​classification discussion to solve the problem of whether the slope exists in parallel and perpendicular problems.

[Example] (1) It is known that two straight lines l1: x+m2y+6=0, l2: (m-2)x+3my+2m=0, if l1∥l2, find the value of the real number m;

(2) It is known that two straight lines l1: ax+2y+6=0 and l2: x+(a-1)y+(a2-1)=0. If l1⊥l2, find the value of the real number a.

[Idea Analysis] Use the condition that the two straight lines are parallel and perpendicular to solve the problem, and pay attention to the situation where the slope is 0 or the slope does not exist.

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Update Time: 2024-11-26

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