Answer: 5x - 3y + 5k = -66
Explanation:
Since AB = AD, we know that triangle ABD is an isosceles triangle, and therefore the line joining the midpoints of AD and AB is perpendicular to BD and bisects AC.
The midpoint of BD is ((k+8)/2, (1+7)/2) = ((k+8)/2, 4)
The gradient of the line 5y - 3x = 6 is 3/5, so the gradient of the line L is also 3/5 since it is parallel to this line.
Since the line L passes through B and D, its equation is given by:
y - 1 = (3/5)(x - k)
y = (3/5)x + (2/5)k + 1
Since AC is perpendicular to BD, it has gradient -5/3.
Therefore, the equation of AC can be found by using the point-slope form of the equation of a straight line and the fact that it passes through the midpoint of BD:
y - 4 = (-5/3)(x - (k+8)/2)
y = (-5/3)x + (5/3)k + (22/3)
Multiplying through by 3, we get:
-5x + 5k + 66 = 3y
5x - 3y + 5k = -66