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ĐH KD & CN Hà Nội 28/09/2022

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Parametric equation. How to write parametric equations of a line

Parametric equations, canonical equations, direction vectors, slopes, … are the key knowledge in the Math 10 program, Geometry subject. In order to help students better understand the theory of parametric equations and how to write parametric equations of straight lines, Trinh Hoai Duc High School shared the following article.

I. HOW TO WRITE PARAMETAL EQUATION OF SUPER SURPRISE LINES

1. What is a parametric equation?

Parametric equations defined by a system of functions of one or more independent variables are called parameters. Parametric equations are often used to represent the coordinates of points on geometric features such as curves or surfaces, which are then called parametric or parametric representations.

For example, the equation:

${displaystyle {begin{aligned}x&=cos ty&=sin tend{aligned}}}$

is the parametric representation of the unit circle, where t is the parameter.

2. Direction vector

– Given a straight line d, vector

is called the direction vector of the line d if the price is parallel or coincident with d.

– is the direction vector of the line d, then is also the direction vector of the line d.

– The direction vector and the normal vector are perpendicular to each other or in other words, the direction vector of d is then the normal vector is .

3. How to write the parametric equation of the line

– Parametric equation of the line passing through the point A(x₀; y₀) take as direction vector, we have:

$Bleft( x,y right)in dLeftrightarrow overrightarrow{AB}=toverrightarrow{u}Leftrightarrow left{ begin{matrix} x-{{x}_{0}}=at y-{{y}_{0}} =bt end{matrix} right.$

$Leftrightarrow left{ begin{matrix} x={{x}_{0}}+at y={{y}_{0}}+bt end{matrix} right., {{a}^{2} }+{{b}^{2}}ne 0, believe mathbb{R}$

– The line d passes through the point A (x₀; y₀), take is the direction vector, the canonical equation of the line is with (a; b 0)

– If

4. Example:

Example 1: Write an equation for the line d that passes through M( -2; 3) and has VTCP u→ = (1; -4) .

A. How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math B. C. D.

The answer

The line (d) passes through M(-2; 3) and has VTCP u→ = (1; -4) should have the equation

How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math

Choose B.

Example 2. The line d passes through the point M( 1; -2) and has a direction vector u→ = (3; 5) has the parametric equation as:

A. d: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math B. d: C. d: D. d:

The answer

Straight line d: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math

⇒ Parametric equation of line d: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math (t CHEAP)

Choose B.

II. EXERCISES ON PARAMETERS OF LINES IN THE ROOM

1. Exercises with answers

Lesson 1: The line passing through two points A(3; -7) and B( 1; -7) has the parametric equation:

A. How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math B. C. D.

The answer

+ We have the line AB: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math

⇒ Equation AB: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math

+ For t = – 3 we get: M( 0; -7) belongs to the line AB.

⇒ AB: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math

⇒ Parametric equation of AB : How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math

Choose A.

Lesson 2: The line d passes through the origin O and has a direction vector u→ = (-1; 2) has the parametric equation as:

A. d: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math B. d: C. d: D. d:

Answer:

Straight line d:
How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math

⇒ Parametric equation d: How to write parametric equations, canonical equations of extremely cool lines - Grade 10 math (t CHEAP)

Answer:

Lesson 3: Give 3 points A(-2; 1), B(-1; 5), C(-2; -3)

a. Write parametric equations AB and AC.

b. Write parametric equations for the midline of BC.

c. Write an equation of the line parallel to AB and passing through the midpoint of BC.

Solution guide

a. Equation of line AB receives make normal vector

The parametric equation AB is: and the canonical equation for d is: $frac{x+2}{1}=frac{y-1}{4}$

Similar to the line AC whose parametric equation is:

b. The perpendicular bisector of BC passes through the midpoint of BC and receives make a normal vector. So the direction vector of the perpendicular is

Let M be the mid point of BC then:

The parametric equation of the perpendicular BC is:

c. Since the line d is parallel to AB, so

According to sentence b, the midpoint of BC is

So the parametric equation of d is:

Lesson 4:

Write the equation of the line y = ax + b known

a) Pass through 2 points A(-3.2), B (5,-4). Calculate the area of the triangle formed by the line and 2 coordinate axes.

b) Pass through A (3,1) parallel to the line y = -2x + m -1.

Solution guide

a. Let the general equation be: y = ax + b

Since the equation of the line passes through 2 points A and B, we have:

$left{ {begin{array}{*{20}{c}} {2 = - 3a + b} { - 4 = 5a + b} end{array}} right. Rightarrow left( {a;b} right) = left( { - frac{3}{4}; - frac{1}{4}} right)$

So the general PT to look for is: $y = - frac{3}{4}x - frac{1}{4}$

The intersection of the line with the Ox axis is: $y = 0 Rightarrow x = - frac{1}{3} Rightarrow Aleft( { - frac{1}{3};0} right)$

$Rightarrow overrightarrow {OA} = left( { - frac{1}{3};0} right) Rightarrow left| {overrightarrow {OA} } right| = frac{1}{3}$

The intersection of the line with the axis Oy is: $x = 0 Rightarrow y = - frac{1}{4} Rightarrow Bleft( {0; - frac{1}{4}} right)$

$Rightarrow overrightarrow {OB} = left( {0; - frac{1}{4}} right) Rightarrow left| {overrightarrow {OB} } right| = frac{1}{4}$

$Rightarrow {S_{OAB}} = frac{1}{2}.OA.OB = frac{1}{2}.frac{1}{3}.frac{1}{4} = frac{1}{{ 24}}$

b. Let the general equation be: y = ax + b

Since the line is parallel to y = -2x + m -1

a = -2

The equation of the line becomes y = -2x + b

Which line passes through point A(3; 1)

1 = 3.(-2) + b

b = 7

So the general equation is: y = -2x + 7

Lesson 5: Write parametric equations, canonical equations of the line d in the following cases:

a. The line d passes through 2 points A(-1;1), B(2; -1).

b. The line d passes through the origin and is parallel to the line

Solution guide

a. We have a straight line d that goes through 2 points A and B, so d gets make direction vectors.

The parametric equation of the line d is: $left{ begin{matrix} x=-1+3t y=1-2t end{matrix} right.$

The canonical equation of the straight line is: $frac{x+1}{3}=frac{y-1}{-2}$

b. We have d parallel to

The parametric equation of the line d is:

The canonical equation for d is:

Lesson 6:Write parametric equations, canonical equations of the line d in the following cases:

a. The equation passing through point A(1; 2) takes make a normal vector.

b. The equation passing through the point B(0; 1) is perpendicular to the line y = 2x + 1.

c. The equation is parallel to the line 4x + 3y – 1 = 0 and passes through the point M( 0, 1).

Solution guide

a. Call the point M(x, y) of d, we have:

$overrightarrow{AM}=toverrightarrow{u}Leftrightarrow left{ begin{matrix} x-1=t y-2=-t end{matrix}Leftrightarrow right.$ $left{ begin{matrix} x=1+t y=2-t end{matrix} right.$

The canonical equation is:

b. We have a line y = 2x + 1 with a normal vector

Since the line d is perpendicular to the line y = 2x + 1, VTPT of y = 2x + 1 is VTCP of d $Rightarrow overrightarrow{n}=overrightarrow{u}=left( -2.1 right)$

We have the parametric equation of d as:

_{2. Extra practice}

Lesson 1:

1. Give 3 points A(-4;1), B(0,2), C(3;-1).

a) Write parametric equations of the lines AB, BC, CA.

b) Let M be the mid point of BC. Write a parametric equation for the line AM.

2. Let ABC be a triangle with A(1;4); B(-9;0); C(7;1)

a) Write parametric equations of lines AB, BC, CA.

b) Write parametric equations for the median of triangle ABC.

Lesson 2: Given 2 straight lines $left( {{d_1}} right):left{ {begin{array}{*{20}{c}} {x = 4 - 3t} {y = - 1 + 2t} end{array}} right.; left( {{d_2}} right):x + 2y - 1 = 0$

a) Find the coordinates of the intersection point A of d_first and d₂

b) Write parametric and general equations for:

+ The line passes through A and is perpendicular to d_first

+ The line passes through A and is parallel to d₂

Lesson 3: Let ABC be a triangle with A(-2; 1), B(-1; 5), C(2; 3)

a. Write parametric equations for sides AB, BC, AC.

b. Write an equation for the medians AM, CP, where M and P are the midpoints of sides BC and AB, respectively.

c. Write the parametric equation for the altitude AH.

d. Write the equation of the line that passes through A and is parallel to BC.

e. The line passes through B and is perpendicular to y = 2x – 3.

Lesson 4: Write parametric equations, canonical equations (if any) in the following cases:

a. The line passes through 2 points A(-2; 0), B(1; 3).

b. The line passing through M(3; -2) is parallel to the line 2x + 5y – 4 = 0.

c. The line with slope k = 1 passes through the point D(-1; -1).

d. The line d passes through the origin and is perpendicular to the line x – y – 1 = 0.

So we have introduced you to the theory of parametric equations and how to write parametric equations of extremely cool lines. Hopefully, after sharing the same article, you have a better grasp of this extremely important piece of knowledge. see more how to find the direction vector of the line at this link!

Posted by: Trinh Hoai Duc High School

Category: General Knowledge

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Parametric equation.  How to write parametric equations of a line
Parametric equations, canonical equations, direction vectors, slopes, ... are the key knowledge in the Math 10 program, Geometry subject.  In order to help students better understand the theory of parametric equations and how to write parametric equations of straight lines, Trinh Hoai Duc High School shared the following article.
I. HOW TO WRITE PARAMETAL EQUATION OF SUPER SURPRISE LINES
1. What is a parametric equation?
Parametric equations defined by a system of functions of one or more independent variables are called parameters.  Parametric equations are often used to represent the coordinates of points on geometric features such as curves or surfaces, which are then called parametric or parametric representations.
For example, the equation:
 ${displaystyle {begin{aligned}x&=cos ty&=sin tend{aligned}}}$ 
is the parametric representation of the unit circle, where t is the parameter.





2. Direction vector
– Given a straight line d, vector 





 is called the direction vector of the line d if the price is parallel or coincident with d.
–  is the direction vector of the line d, then  is also the direction vector of the line d.
– The direction vector and the normal vector are perpendicular to each other or in other words, the direction vector of d is  then the normal vector is .
3. How to write the parametric equation of the line
– Parametric equation of the line passing through the point A(x₀;  y₀) take  as direction vector, we have:


– The line d passes through the point A (x₀;  y₀), take  is the direction vector, the canonical equation of the line is  with (a; b 0)
- If 
4. Example:
Example 1: Write an equation for the line d that passes through M( -2; 3) and has VTCP u→ = (1; -4) .
A.     B.     C.     D. 
The answer
The line (d) passes through M(-2; 3) and has VTCP u→ = (1; -4) should have the equation

Choose B.
Example 2. The line d passes through the point M( 1; -2) and has a direction vector u→ = (3; 5) has the parametric equation as:
A. d:     B. d:     C. d:     D. d: 

The answer
Straight line d: 
⇒ Parametric equation of line d:  (t CHEAP)
Choose B.
II.  EXERCISES ON PARAMETERS OF LINES IN THE ROOM
1. Exercises with answers
Lesson 1: The line passing through two points A(3; -7) and B( 1; -7) has the parametric equation:
A.     B.     C.     D. 
The answer
+ We have the line AB: 
⇒ Equation AB: 
+ For t = - 3 we get: M( 0; -7) belongs to the line AB.
⇒ AB: 
⇒ Parametric equation of AB : 
Choose A.
Lesson 2: The line d passes through the origin O and has a direction vector u→ = (-1; 2) has the parametric equation as:
A. d:     B. d:     C. d:     D. d: 
Answer:
Straight line d:

⇒ Parametric equation d:  (t CHEAP)
Answer:
Lesson 3: Give 3 points A(-2; 1), B(-1; 5), C(-2; -3)
a.  Write parametric equations AB and AC.
b.  Write parametric equations for the midline of BC.
c.  Write an equation of the line parallel to AB and passing through the midpoint of BC.
Solution guide
a.  Equation of line AB receives  make normal vector
The parametric equation AB is:  and the canonical equation for d is: 
Similar to the line AC whose parametric equation is: 
b.  The perpendicular bisector of BC passes through the midpoint of BC and receives  make a normal vector.  So the direction vector of the perpendicular is 
Let M be the mid point of BC then: 
The parametric equation of the perpendicular BC is: 
c.  Since the line d is parallel to AB, so 
According to sentence b, the midpoint of BC is 
So the parametric equation of d is: 
Lesson 4:
Write the equation of the line y = ax + b known
a) Pass through 2 points A(-3.2), B (5,-4).  Calculate the area of the triangle formed by the line and 2 coordinate axes.
b) Pass through A (3,1) parallel to the line y = -2x + m -1.
Solution guide
a.  Let the general equation be: y = ax + b
Since the equation of the line passes through 2 points A and B, we have:

So the general PT to look for is: 
The intersection of the line with the Ox axis is: 

The intersection of the line with the axis Oy is: 


b.  Let the general equation be: y = ax + b
Since the line is parallel to y = -2x + m -1
a = -2
The equation of the line becomes y = -2x + b
Which line passes through point A(3; 1)
1 = 3.(-2) + b
b = 7
So the general equation is: y = -2x + 7
Lesson 5: Write parametric equations, canonical equations of the line d in the following cases:
a.  The line d passes through 2 points A(-1;1), B(2; -1).
b.  The line d passes through the origin and is parallel to the line 
Solution guide
a.  We have a straight line d that goes through 2 points A and B, so d gets  make direction vectors.
The parametric equation of the line d is:  
The canonical equation of the straight line is: 
b.  We have d parallel to 
The parametric equation of the line d is: 
The canonical equation for d is: 
Lesson 6:Write parametric equations, canonical equations of the line d in the following cases:
a.  The equation passing through point A(1; 2) takes  make a normal vector.
b.  The equation passing through the point B(0; 1) is perpendicular to the line y = 2x + 1.
c.  The equation is parallel to the line 4x + 3y – 1 = 0 and passes through the point M( 0, 1).
Solution guide
a.  Call the point M(x, y) of d, we have:

The canonical equation is: 
b.  We have a line y = 2x + 1 with a normal vector 
Since the line d is perpendicular to the line y = 2x + 1, VTPT  of y = 2x + 1 is VTCP  of d 
We have the parametric equation of d as: 
_{2. Extra practice}
Lesson 1:
1. Give 3 points A(-4;1), B(0,2), C(3;-1).
a) Write parametric equations of the lines AB, BC, CA.
b) Let M be the mid point of BC.  Write a parametric equation for the line AM.
2. Let ABC be a triangle with A(1;4);  B(-9;0);  C(7;1)
a) Write parametric equations of lines AB, BC, CA.
b) Write parametric equations for the median of triangle ABC.
Lesson 2: Given 2 straight lines 
a) Find the coordinates of the intersection point A of d_first and d₂
b) Write parametric and general equations for:
+ The line passes through A and is perpendicular to d_first
+ The line passes through A and is parallel to d₂
Lesson 3: Let ABC be a triangle with A(-2; 1), B(-1; 5), C(2; 3)
a.  Write parametric equations for sides AB, BC, AC.
b.  Write an equation for the medians AM, CP, where M and P are the midpoints of sides BC and AB, respectively.
c.  Write the parametric equation for the altitude AH.
d.  Write the equation of the line that passes through A and is parallel to BC.
e.  The line passes through B and is perpendicular to y = 2x – 3.
Lesson 4: Write parametric equations, canonical equations (if any) in the following cases:
a.  The line passes through 2 points A(-2; 0), B(1; 3).
b.  The line passing through M(3; -2) is parallel to the line 2x + 5y – 4 = 0.
c.  The line with slope k = 1 passes through the point D(-1; -1).
d.  The line d passes through the origin and is perpendicular to the line x - y - 1 = 0.
So we have introduced you to the theory of parametric equations and how to write parametric equations of extremely cool lines.  Hopefully, after sharing the same article, you have a better grasp of this extremely important piece of knowledge.  see more how to find the direction vector of the line at this link!
Posted by: Trinh Hoai Duc High School
Category: General Knowledge

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