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

$Answer \: refer \: to \: the \: attachment \:$

Answer 2

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

Given that

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

Let us assume that

• Slope of other line is m.

We know that

Angle x between two lines having slope m and M is given by

$\red{\rm :\longmapsto\:\boxed{\tt{ \: tanx \: = \: \bigg | \frac{M - m}{1 + Mm} \bigg| \: }}}$

So, here

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

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

and

$\red{\rm :\longmapsto\:m = m \: }$

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

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

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

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

Case - 1

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

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

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

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

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

Case - 2

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

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

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

$\red{\rm \implies\:\boxed{\tt{ m \: = \: 3 \: }}}$

Hence,

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

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More to Know

1. Slope of a line is defined as tangent of the angle which a line makes with the positive direction of x axis measured in anti-clockwise direction.

If angle is obtuse, m < 0

If angle is acute, m > 0

If angle is 0°, m = 0

If angle is 90°, m is not defined.

2. Two lines having slope m and M are parallel iff m = M

3. Two lines having slope m and M are perpendicular iff Mm = - 1

4. If line is parallel to x- axis, slope of line is 0

5. If line is parallel to y - axis, slope is not defined.