The straight line ax+by+c=0 bisects an angle between a pair of straight lines of which one is lx+my+n=0. Find the equation of the other line of the pair.
Hint: Try using the fact that the angle denoted by cos(0.5x) = <d1,d2> where d1, d2 are unit direction vectors for the lines and that the direction vector for the third line is rotated x radians from the line lx + my + n = 0 in the plane that is formed by the existing two lines.
Geometrically, the line you seek is very easily described. It's just the reflection of the second line about the first. Reasonable dynamic software allows you to do the reflection with just a couple of operations. However, you wanted the equation of the reflected line. It's kind of a mess algebraically, but I got started with it and kept going. Here's the result in three attached images: (I changed your second equation to read dx + ey +f = 0; the third image I made to test out my algebra
Since we have the equations of lines in general form the expressions would become quite big. The algorithm for doing such questions would be as follows.
1.Find the point of intersection of two lines.
2. Since one of the given lines is the angle bisector implies that the angles on both the sides of bisector are equal.
3. Let the slope of the required line be 'm'.
4. Equate the angle on either side of bisector using the expression for angle between two lines: (m1 - m2)/(1-m1m2)
5. Get the value of m and then find the equation of line through the point of intersection with the given slope.
Alternatively you can proceed by writing the general equation of a straight line passing through the point of intersection of given two lines which is given by
(ax+by+c) + K ( lx+my+n) = 0 where k is not zero.
Now find the slope and equate it to the slope of required line found by equating angle as explained earlier. That will give the value of k. OK