Question

Here's a question ~

Two lines passing through the point (2 , 3) intersect each other at an angle of 60° . If slope of one line is 2, then find the equation of the other line ~

note : there are two possible slopes !

Need Proper Explanation ~​

Answer 1

`\sf\longmapsto \: ( √(3) - 2)x + (2 √(3 ) + 1)y = - 1 + 8 √(3)`

Explanation:

It is given that –

Slope of the first line is–

`\sf\longmapsto \: m_(1) = 2`

Let,

the slope of the Another line be –

`\sf\longmapsto \: m _(2)`

Now,

The angle between the two lines is 60°.

Let's start solving!

`\sf\longmapsto \: \tan(60°) | \frac{ m_(1) - m_ {2} }{1 + m_(1) m_ {2}} |`

`\sf\longmapsto \: √(3) = | (2 -m_(2) )/(1 + 2m_(2)) |`

`\sf\longmapsto \: √(3) = ± \: \: ( (2 -m_(2) )/(1 + 2m_(2)) )`

`\sf\longmapsto \: √(3) = (2 -m_(2) )/(1 + 2m_(2))`

`\sf\longmapsto \: √(3) (1 + 2m_(2)) = 2 - m_(2)`

`\sf\longmapsto \: √(3) + 2 √(3) m_(2) + m_(2) = 2`

`\sf\longmapsto \: m_(2) = (2 - √(3) )/((2 √(3) + 1) )`

The equation of line passing through point (2,3) and having a slope of –

`\sf\longmapsto \: m_(2) = ((2 - √(3) ))/(2 √(3) + 1 )`

is–

`\sf\longmapsto \: (y - 3) = (2 - √(3) )/(2 √(3) + 1 ) (x - 2)`

`\sf\longmapsto \: ( 2√(3) + 1)y - 3(2 √(3) + 1) = (2 - √(3) )x - 2(2 - √(3)`

`\sf\longmapsto \: ( √(3) - 2)x + (2 √(3 ) + 1)y = - 1 + 8 √(3)`

Hence the equation of the other line is -

`\sf\longmapsto \: ( √(3) - 2)x + (2 √(3 ) + 1)y = - 1 + 8 √(3)`