Find the joint equation of the lines passing through the origin, each of which making angle of measure `15^(@)` with the line `x - y = 0`.

Explanation

Let m be the slope of the line `x-y = 0.`
`therefore m = 1 = 45^(@)`
`therefore` the line `x - y = 0` makes an angle of `45^(@)` with the positive side of X-axis. Let OA and OB be the two lines through the origin, each making an angle of `15^(@)` with the line `x-y =0.`

`therefore` OA and OB make an angle of `30^(@)` and `60^(@)` with the positive side of X-axis
`therefore` slope of OA ` = tan 30^(@) = (1)/(sqrt(3))`
`therefore` equation of the line oA is `y = (1)/(sqrt(3))x, i.e.,sqrt(3)x - y = 0`
`therefore` required joint equation of the lines is
`(x-sqrt(3)y)(sqrt(3)x-y) = 0`
`therefore sqrt(3)x^(2) - xy - 3xy + sqrt(3)y^(2) = 0`
`therefore sqrt(3)x^(2) - 4xy + sqrt(3)y^(2) = 0.`