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(C) Step $1$: Translation of axes. The original coordinates are $(x, y) = (2, 3)$ and the origin is shifted to $(h, k) = (3, 2)$. The new coordinates

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$\sqrt{2}$

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(C) Step $1$: Translation of axes. The original coordinates are $(x, y) = (2, 3)$ and the origin is shifted to $(h, k) = (3, 2)$. The new coordinates $(a, b)$ are given by $a = x - h = 2 - 3 = -1$ and $b = y - k = 3 - 2 = 1$. So,$(a, b) = (-1, 1)$.Step $2$: Rotation of axes. The point $(a, b) = (-1, 1)$ is rotated by $	heta = \frac{\pi}{4}$ anti-clockwise. The new coordinates $(c, d)$ are given by $c = a \cos 	heta + b \sin 	heta$ and $d = -a \sin 	heta + b \cos 	heta$.Step $3$: Calculate $c$ and $d$. Since $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$,we have $c = (-1)(\frac{1}{\sqrt{2}}) + (1)(\frac{1}{\sqrt{2}}) = 0$ and $d = -(-1)(\frac{1}{\sqrt{2}}) + (1)(\frac{1}{\sqrt{2}}) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$.Step $4$: Calculate $d - c = \sqrt{2} - 0 = \sqrt{2}$.

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