Question

find the target of the acute angle between the following lines:2x+y=2 and 3x+y=-1?

Answer 1

Appropriate Question :-

Find the tangent of the acute angle between the following lines : 2x + y = 2 and 3x + y = - 1.

$\large\underline{\sf{Solution-}}$

Given equation of lines are

$\rm \: 2x + y = 2$

and

$\rm \: 3x + y = - 1$

Consider first equation

$\rm \: 2x + y = 2$

can be rewritten as

$\rm \: y = - 2x + 2$

Its the equation of line of the form y = mx + c, where m represents the slope of line.

So, slope of the line is

$\rm\implies \:m_1 = - 2$

Now, Consider second equation of line

$\rm \: 3x + y = - 1$

$\rm \: y = - 3x - 1$

So, slope of this line is

$\rm\implies \:m_2 = - 3$

Now, We know that,

The angle between two lines having slope m and M is given by

$\rm \: tan \theta \: = \: \bigg |\dfrac{M - m}{1 + Mm} \bigg|$

$\rm \: where \: \theta \: is \: the \: angle \: between \: two \: lines$

Now, here

$\rm \: M = m_1 = - 2$

and

$\rm \: m = m_2 = - 3$

So, on substituting the values, we get

$\rm \: tan \theta \: = \: \bigg |\dfrac{ - 2 - ( - 3)}{1 +( - 2)( - 3)} \bigg|$

$\rm \: tan \theta \: = \: \bigg |\dfrac{ - 2 + 3}{1 +6} \bigg|$

$\rm \: tan \theta \: = \: \bigg |\dfrac{1}{7} \bigg|$

$\rm\implies \:\theta = {tan}^{ - 1}\bigg(\dfrac{1}{7} \bigg)$

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Additional Information :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.

Answer 2

Answer:

Appropriate Question :-

Find the tangent of the acute angle between the following lines : 2x + y = 2 and 3x + y = - 1.

Given equation of lines are

and

Consider first equation

can be rewritten as

Its the equation of line of the form y = mx + c, where m represents the slope of line.

So, slope of the line is

Now, Consider second equation of line

So, slope of this line is

Now, We know that,

The angle between two lines having slope m and M is given by

Now, here

and

So, on substituting the values, we get

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Additional Information :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.