Question

If the angle between two lines is 45 ° and the slope of one of the line is 1/2 find the slope of the other line.

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Answer 1

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

Given that

• The angle between two lines is 45 ° and the slope of one of the line is 1/2.

Let assume that

• The slope of other line be m.

We know,

Angle p between two lines having slope m and M respectively is given

$\red{\rm :\longmapsto\:\boxed{\tt{ tan \: p = \bigg | \dfrac{M - m}{1 + Mm} \bigg| }}}$

Here,

$\red{\rm :\longmapsto\:p = 45\degree}$

$\red{\rm :\longmapsto\:M = \dfrac{1}{2}}$

So, on substituting the values in above formula, we get

$\rm :\longmapsto\:tan45\degree = \bigg |\dfrac{\dfrac{1}{2} - m}{1 + \dfrac{1}{2} m} \bigg|$

$\rm :\longmapsto\:1 = \bigg |\dfrac{1 - 2m}{2 + m} \bigg|$

$\rm :\longmapsto\: \pm \: 1 \: = \: \dfrac{1 - 2m}{m + 2}$

Case 1

$\rm :\longmapsto\: 1 \: = \: \dfrac{1 - 2m}{m + 2}$

$\rm :\longmapsto\:m + 2 = 1 - 2m$

$\rm :\longmapsto\:m + 2m = 1 - 2$

$\rm :\longmapsto\:3m = - 1$

$\bf\implies \:\boxed{\tt{ \: m \: = \: - \: \frac{1}{3} \: }}$

Case 2

$\rm :\longmapsto\: - \: 1 \: = \: \dfrac{1 - 2m}{m + 2}$

$\rm :\longmapsto\: - m - 2 = 1 - 2m$

$\rm :\longmapsto\: - m + 2m = 1 + 2$

$\bf\implies \:\boxed{\tt{ \: m \: = \: 3 \: }}$

Hence,

$\bf\implies \:\boxed{\tt{ \: m \: = \: 3 \: \: \: or \: \: \: - \: \frac{1}{3} \: }}$

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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 x = a.

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

Easy explanation:

• If angle between two lines with slopes m1 and m2 is α then tan α = |(m1-m2)/(1+m1*m2)|

tan 45° = $|\frac{m-1/4}{1+m/4}|$

=> $\frac{4m-1}{m+4}=-1$

=>- m-4 = 4m-1 => 5m = -3

=> m = -3/5.

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