A straight line through the point \((2,2)\) intersects the line \(\sqrt{3 x}+y=0\) and \(\sqrt{3 x}-y=0\) at the points \(\mathrm{A}\) and \(B\). The equation to the line \(\mathrm{AB}\) so that the triangle \(\mathrm{OAB}\) is equilateral is (a) \(x=2\) (b) \(\mathrm{y}=2\) (c) \(\mathrm{x}+\mathrm{y}=4\) (d) none

Since both lines intersect at point (2,2), we can substitute the x and y coordinates of this point into the equations for the lines: For the line \(\sqrt{3 x}+y=0\), substitute (2,2): \( \sqrt{3(2)} + 2 = 0 \) For the line \(\sqrt{3 x}-y=0\), substitute (2,2): \( \sqrt{3(2)} - 2 = 0 \) Observe that both equations are satisfied, so the points A and B are the intersections of the lines with point (2,2).

We will now test each of the given options to see which one forms an equilateral triangle OAB. (a) \(x=2\) This equation represents a vertical line passing through the point (2,2). Since it is impossible to form an equilateral triangle on a vertical line, this is not the correct option. (b) \(y=2\) This equation represents a horizontal line passing through the point (2,2). In this case, since points A and B lie on the lines \(\sqrt{3 x}+y=0\) and \(\sqrt{3 x}-y=0\), the points A and B will have the same y-coordinate (2). Also, the distance between the point (0,0) and A≠the distance between the point (0,0) and B. So, it is not possible to form an equilateral triangle OAB on this line. (c) \(x+y=4\) This line passes through the point (2,2), and we need to find the coordinates of points A and B on this line. Substitute the equation of line \(x+y=4\) into the equation of lines \(\sqrt{3 x}+y=0\) and \(\sqrt{3 x}-y=0\): For the line \(\sqrt{3 x}+y=0\): \( \sqrt{3(4-y)} + y = 0 \) For the line \(\sqrt{3 x}-y=0\): \( \sqrt{3(4-y)} - y = 0 \) Solving these equations, we find that neither of the equations has a real solution. This indicates that the intersection of line \(x+y=4\) with the lines \(\sqrt{3 x}+y=0\) and \(\sqrt{3 x}-y=0\) does not exist. Hence, this option is not correct either. (d) None Since none of the given options form an equilateral triangle, the correct answer is (d) None.