### Related Formula y - y_1 = tan(theta)(x - x_1)$y - y_1 = \tan(\theta)(x - x_1)$ ### Core Logic

The initial line makes an angle of 30^circ$30^{\circ}$ with the positive x-axis. It is rotated clockwise by 15^circ$15^{\circ}$ about the point A(9, 0)$A(9, 0)$. The new angle made by the line with the positive direction of the x-axis is 30^circ - 15^circ = 15^circ$30^{\circ} - 15^{\circ} = 15^{\circ}$. ### Step 1: Equation of the new line The equation of the line passing through A(9, 0)$A(9, 0)$ with a slope of tan 15^circ$\tan 15^{\circ}$ is: textEq^n: y - 0 = tan 15^circ (x - 9)$\text{Eq}^n: y - 0 = \tan 15^{\circ} (x - 9)$ We know that tan 15^circ = 2 - sqrt3$\tan 15^{\circ} = 2 - \sqrt{3}$. y = (2 - sqrt3) (x - 9)$y = (2 - \sqrt{3}) (x - 9)$ ### Step 2: Rearranging to match options Dividing by (2 - sqrt3)$(2 - \sqrt{3})$: fracy2 - sqrt3 = x - 9$\frac{y}{2 - \sqrt{3}} = x - 9$ Notice that 2 - sqrt3 = -(sqrt3 - 2)$2 - \sqrt{3} = -(\sqrt{3} - 2)$. Thus: frac-ysqrt3 - 2 = x - 9$\frac{-y}{\sqrt{3} - 2} = x - 9$ fracysqrt3 - 2 + x = 9$\frac{y}{\sqrt{3} - 2} + x = 9$ ### Pattern Recognition A clockwise rotation decreases the angle of inclination. Calculate the new angle, find its tangent, and carefully algebraicize the denominator to match the given option forms. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Straight Lines