If origin is shifted to the point (2,3) and the axes are rotated through an angle `pi//4` in the anticlokwise direction, then find the coordinates of the point (7, 11) in the new system of coordinates.

Answer

To find the coordinates of the point (7, 11) in the new system of coordinates after shifting the origin to (2, 3) and rotating the axes through an angle of π/4 in the anticlockwise direction, we will follow these steps:

### Step 1: Shift the Origin
We first need to shift the origin from (0, 0) to (2, 3). To do this, we will subtract the coordinates of the new origin from the coordinates of the point (7, 11).

- New coordinates after shifting the origin:
\[ (x', y') = (x - 2, y - 3) \]
where \( (x, y) = (7, 11) \).

Calculating:
\[ x' = 7 - 2 = 5 \]
\[ y' = 11 - 3 = 8 \]

So, the new coordinates after shifting the origin are \( (5, 8) \).

### Step 2: Rotate the Axes
Next, we need to rotate the axes through an angle of π/4. The transformation for rotating a point \((x', y')\) through an angle θ is given by:
\[ x'' = x' \cos(\theta) - y' \sin(\theta) \]
\[ y'' = x' \sin(\theta) + y' \cos(\theta) \]
For θ = π/4, we have:
\[ \cos(\pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2} \]

Substituting the values:
\[ x'' = 5 \cdot \frac{\sqrt{2}}{2} - 8 \cdot \frac{\sqrt{2}}{2} \]
\[ y'' = 5 \cdot \frac{\sqrt{2}}{2} + 8 \cdot \frac{\sqrt{2}}{2} \]

Calculating:
\[ x'' = \frac{\sqrt{2}}{2} (5 - 8) = \frac{\sqrt{2}}{2} (-3) = -\frac{3\sqrt{2}}{2} \]
\[ y'' = \frac{\sqrt{2}}{2} (5 + 8) = \frac{\sqrt{2}}{2} (13) = \frac{13\sqrt{2}}{2} \]

Thus, the coordinates of the point (7, 11) in the new system of coordinates after shifting the origin and rotating the axes are:
\[ \left(-\frac{3\sqrt{2}}{2}, \frac{13\sqrt{2}}{2}\right) \]

### Summary of the Solution
The coordinates of the point (7, 11) in the new system of coordinates after shifting the origin to (2, 3) and rotating the axes through an angle of π/4 are:
\[ \left(-\frac{3\sqrt{2}}{2}, \frac{13\sqrt{2}}{2}\right) \]