Straight Lines and Pair of Straight Lines

Let the original coordinates of the point $P$ be $(x_0, y_0) = (4,1)$.We will apply the transformations in succession and find the coordinates of $P$ after each step.Step (i): Origin is shifted to the point $(1,6)$ by translation of axes.When the origin is shifted to $(h, k)$, the new coordinates $(x', y')$ of a point $(x, y)$ are given by the formulas:$\; x' = x - h$$\; y' = y - k$In this case, $(x, y) = (4,1)$ and $(h, k) = (1,6)$.So, the coordinates of $P$ after this transformation, let's call them ( $x_1, y_1$ ), are:$\; x_1 = 4 - 1 = 3$$\; y_1 = 1 - 6 = -5$The point $P$ is now at $(3,-5)$ with respect to the new coordinate system.Step (ii): Translation through a distance of $2$ units along the positive direction of $X$-axis.This implies a translation of the point itself within the current coordinate system (established after step (i)). A translation of a point $(x, y)$ by $a$ units along the positive X-axis results in new coordinates ( $x + a, y$ ).Here, the point is $(x_1, y_1) = (3,-5)$, and the translation distance is $a = 2$.So, the coordinates of $P$ after this transformation, let's call them $(x_2, y_2)$, are:$\; x_2 = x_1 + 2 = 3 + 2 = 5$$\; y_2 = y_1 = -5$The point $P$ is now at $(5,-5)$.Step (iii): Rotation of axes through an angle of $90^{\circ}$ in the positive direction.This means the coordinate axes (with respect to which $P$ is currently at $(5,-5)$ ) are rotated by an angle $\theta = 90^{\circ}$ counter-clockwise.If the old coordinates of a point are $(x, y)$ and the axes are rotated by an angle $\theta$, the new coordinates $(x', y')$ are given by:$\; x' = x \cos \theta + y \sin \theta$$\; y' = -x \sin \theta + y \cos \theta$In this case, the point is $(x_2, y_2) = (5,-5)$ and $\theta = 90^{\circ}$.$\cos 90^{\circ} = 0$$\sin 90^{\circ} = 1$So, the final coordinates of $P$, let's call them $(x_3, y_3)$, are:$\; x_3 = 5 \cos 90^{\circ} + (-5) \sin 90^{\circ} = 5(0) + (-5)(1) = -5$$\; y_3 = -5 \sin 90^{\circ} + (-5) \cos 90^{\circ} = -5(1) + (-5)(0) = -5$The final position of the point $P$ is $(-5,-5)$.Comparing this with the given options, the final coordinates are $(-5,-5)$, which corresponds to Option C.The final answer is $(-5,-5)$.