Question

If the angle between two lines is π/4 and slope of one line is x and of another line is y such that xy > 0. If y =1/2 then the value of x is​

Answer 1

Given :-

• Angle between two lines is π/4
• Slope of one line is x and of another line is y such that xy > 0
• y = 1/2

To find :-

• Value of x

Solution :-

The relation between the angle between two lines and their slopes is given by,

$\rm \longrightarrow \tan \theta = \left | \dfrac{m_2 - m_1}{1 + m_1.m_2} \right|$

Here,

• θ = Angle between two lines
• m1 and m2 are slopes of given line

$\rm \longrightarrow \tan \left( \dfrac{ \pi}{4} \right) = \left | \dfrac{y - x}{1 + x.y} \right|$

$\rm \longrightarrow1= \left | \dfrac{ \dfrac{1}{2} - x}{1 + \left( x \times \dfrac{1}{2} \right) } \right|$

$\rm \longrightarrow1= \left | \dfrac{ \dfrac{1 - 2x}{2} }{1 + \left( \dfrac{x}{2} \right) } \right|$

$\rm \longrightarrow1= \left | \dfrac{ \dfrac{1 - 2x}{2} }{\dfrac{2 + x}{2} } \right|$

$\rm \longrightarrow1= \left | \dfrac{1 - 2x }{2 + x } \right|$

$\rm \longrightarrow1= \dfrac{ | 1 - 2x | }{ | 2 + x | }$

$\rm \longrightarrow |2 + x| = | 1 - 2x |$

Now, here are two cases,

$\rm \longrightarrow 2 + x = \pm( 1 - 2x )$

For case I :

$\rm \longrightarrow 2 + x =( 1 - 2x )$

$\rm \longrightarrow 2 + x =1 - 2x$

$\rm \longrightarrow 2 - 1 = - 2x - x$

$\rm \longrightarrow 1 = - 3x$

$\rm \longrightarrow - \dfrac{1}{3} = x$

This case is not possible because it is given that product of x and y is greater than 0.

Case II :-

$\rm \longrightarrow 2 + x = - ( 1 - 2x )$

$\rm \longrightarrow 2 + x = - 1 + 2x$

$\rm \longrightarrow 2 +1= - x + 2x$

$\rm \longrightarrow 3= x$

Hence the value of x is 3.