The joint equation of two altitudes of an equilateral triangle is `(sqrt(3)x-y+8-4sqrt(3)) (-sqrt(3)x-y+12 +4sqrt(3)) = 0` The third altitude has the equation

Answer

y=10

Explanation

Clearly, the third altitude will be the bisector of obtuse angle between the given altitudes.
Given two altitudes are :
`sqrt(3)-y+8-4sqrt(3) = 0 " " (1)`
`"and " -sqrt(3)-y+12+4sqrt(3) = 0 " " (2)`
`"We have "a_(1)a_(2) +b_(1)b_(2) = -sqrt(3)sqrt(3)+(-1)(-1) = -2 lt 0`
Therefore, bisector of obtuse angle is
`(sqrt(3)x-y+8-4sqrt(3))/(sqrt(3)+1) = (-sqrt(3)x-y+12+4sqrt(3))/(sqrt(3)+1)`
or y=10