Question

(a,b) are the new coordinates of the point (2,3) after shifting the origin to the point (3,2) by translation of axes. If (c,d) are the new coordinates of the point (a,b) after rotating the axes through an angle \( \frac{\pi}{4} \) about the origin in the anti-clockwise direction, then d-c =

• A. 0
• B. 1
• C. \( \sqrt{2} \)
• D. \( 2\sqrt{2} \)

Hint:

Remember the coordinate transformation formulas. For translation of origin to (h,k): \(x_{old} = x_{new} + h\), \(y_{old} = y_{new} + k\). For rotation by angle \(\theta\): \(x_{new} = x_{old}\cos\theta - y_{old}\sin\theta\), \(y_{new} = x_{old}\sin\theta + y_{old}\cos\theta\).

Answer 1

The Correct Option is C

Step 1: Find the coordinates (a,b) after translation of the origin.
The original coordinates are \((x,y) = (2,3)\). The new origin is \((h,k) = (3,2)\).
The transformation formulas are \( x = a+h \) and \( y = b+k \).
For x: \( 2 = a + 3 \implies a = -1 \).
For y: \( 3 = b + 2 \implies b = 1 \).
So, the new coordinates after translation are \((a,b) = (-1, 1)\).
Step 2: Find the coordinates (c,d) after rotation.
The point to be rotated is \((a,b) = (-1,1)\) by an angle \( \theta = \frac{\pi}{4} \).
The rotation formulas are \( c = a\cos\theta - b\sin\theta \) and \( d = a\sin\theta + b\cos\theta \).
We know \( \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \) and \( \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \).
\( c = (-1)\left(\frac{1}{\sqrt{2}}\right) - (1)\left(\frac{1}{\sqrt{2}}\right) = -\frac{2}{\sqrt{2}} = -\sqrt{2} \).
\( d = (-1)\left(\frac{1}{\sqrt{2}}\right) + (1)\left(\frac{1}{\sqrt{2}}\right) = 0 \).
Step 3: Calculate the required value d-c.
\( d - c = 0 - (-\sqrt{2}) = \sqrt{2} \).