Equation of line which is equally inclined to the axis and passes through a common points of family of lines \(4 a c x+y(a b+b c+c a-a b c)+a b c=0\) (a) \(y-x=(7 / 4)\) (b) \(\mathrm{y}+\mathrm{x}=(7 / 4)\) (c) \(y-x=(1 / 4)\) (d) \(y+x=(1 / 4)\)

Since the line has equal inclination to both axes, it has a 45-degree angle with both axes. To find the slope of this line, we can use the tangent of the angle: Slope, m = \(tan45° = 1\)

Now that we know the slope of a line that has an equal inclination to both axes should be either 1 or -1, we can look at the given options and identify which option has a slope equal to 1 or -1. We can rewrite the equations to be in slope-intercept form (y = mx + b) to find the slopes: (a) y - x = 7/4 => y = x + 7/4 (Slope m = 1) (b) y + x = 7/4 => y = -x + 7/4 (Slope m = -1) (c) y - x = 1/4 => y = x + 1/4 (Slope m = 1) (d) y + x = 1/4 => y = -x + 1/4 (Slope m = -1) Options (a), (b), (c), and (d) have slopes that are either 1 or -1.

We now have four possible lines with the correct slope: (a), (b), (c), and (d). The given family of lines is defined as: \(4 a c x+y(a b+b c+c a-a b c)+a b c=0\) Since the correct answer should intersect this family of lines at a common point, let's find the coordinates of the intersection point with each option. We can substitute the value of y from each option into the equation of this family of lines and set the equation equal to zero. For (a): Substituting y = x + 7/4, \(4 a c x+(x+(7/4)(a b+b c+c a-a b c)+a b c=0\) For (b): Substituting y = -x + 7/4, \(4 a c x+(-x+(7/4)(a b+b c+c a-a b c)+a b c=0\) For (c): Substituting y = x + 1/4, \(4 a c x+(x+(1/4)(a b+b c+c a-a b c)+a b c=0\) For (d): Substituting y = -x + 1/4, \(4 a c x+(-x+(1/4)(a b+b c+c a-a b c)+a b c=0\) By analyzing the substitution results, we find that the line corresponding to option (a) has the same coefficients as that of the given line \(4 a c x+y(a b+b c+c a-a b c)+a b c=0\). Hence, it intersects this family of lines at a common point.

So the equation of the line which is equally inclined to the axis and passes through a common point of the family of lines, as defined by the given equation, is: \(y-x=(7 / 4)\) The correct answer is (a).