Question:
The origin is shifted to the point \((1,1)\) by translation of axes and then the axes are rotated through an angle
\[
\frac{\pi}{3}
\]
about \((1,1)\) in the positive direction. After these two transformations, if the point
\[
P(3,2)
\]
becomes \(P(\alpha,\beta)\), then
\[
4\alpha+2\beta=
\]
The origin is shifted to the point \((1,1)\) by translation of axes and then the axes are rotated through an angle
\[
\frac{\pi}{3}
\]
about \((1,1)\) in the positive direction. After these two transformations, if the point
\[
P(3,2)
\]
becomes \(P(\alpha,\beta)\), then
\[
4\alpha+2\beta=
\]
Show Hint
After translating the origin, first compute the coordinates relative to the new origin. Then apply the standard rotation-of-axes formulas. Many expressions simplify before you need the exact values of the transformed coordinates.
Updated On: Jul 9, 2026
- \(0\)
- \(5\)
- \(-7\)
- \(15\) \bigskip
Show Solution
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The Correct Option is B
Solution and Explanation
Concept:
If the origin is shifted to \((h,k)\), then the new coordinates are
\[
X=x-h,
\qquad
Y=y-k.
\]
If the axes are then rotated through an angle \(\theta\) in the positive direction, the transformed coordinates are
\[
X=\alpha\cos\theta-\beta\sin\theta,
\]
\[
Y=\alpha\sin\theta+\beta\cos\theta.
\]
Step 1: Translate the origin to \((1,1)\). The point is \[ P(3,2). \] After translation, \[ X=3-1=2, \] \[ Y=2-1=1. \] Thus the coordinates relative to the translated axes are \[ (2,1). \]
Step 2: Use the rotation formulas. The axes are rotated through \[ \theta=\frac{\pi}{3}. \] Hence, \[ \cos\theta=\frac12, \qquad \sin\theta=\frac{\sqrt3}{2}. \] Using \[ X=\alpha\cos\theta-\beta\sin\theta, \] \[ Y=\alpha\sin\theta+\beta\cos\theta, \] we get \[ 2=\frac{\alpha}{2}-\frac{\sqrt3}{2}\beta, \] \[ 1=\frac{\sqrt3}{2}\alpha+\frac{\beta}{2}. \] Multiplying by \(2\), \[ 4=\alpha-\sqrt3\,\beta, \] \[ 2=\sqrt3\,\alpha+\beta. \]
Step 3: Solve for \(\alpha\) and \(\beta\). From \[ 4=\alpha-\sqrt3\,\beta, \] \[ \alpha=4+\sqrt3\,\beta. \] Substituting into the second equation, \[ 2=\sqrt3(4+\sqrt3\,\beta)+\beta. \] \[ 2=4\sqrt3+4\beta. \] \[ \beta=\frac{1-2\sqrt3}{2}. \] Therefore, \[ \alpha = 4+\sqrt3\left(\frac{1-2\sqrt3}{2}\right) = 1+\frac{\sqrt3}{2}. \]
Step 4: Find \(4\alpha+2\beta\). \[ 4\alpha+2\beta = 4\left(1+\frac{\sqrt3}{2}\right) + 2\left(\frac{1-2\sqrt3}{2}\right). \] \[ = 4+2\sqrt3+1-2\sqrt3. \] \[ =5. \]
Step 5: Write the final answer. \[ \boxed{5} \]
Step 1: Translate the origin to \((1,1)\). The point is \[ P(3,2). \] After translation, \[ X=3-1=2, \] \[ Y=2-1=1. \] Thus the coordinates relative to the translated axes are \[ (2,1). \]
Step 2: Use the rotation formulas. The axes are rotated through \[ \theta=\frac{\pi}{3}. \] Hence, \[ \cos\theta=\frac12, \qquad \sin\theta=\frac{\sqrt3}{2}. \] Using \[ X=\alpha\cos\theta-\beta\sin\theta, \] \[ Y=\alpha\sin\theta+\beta\cos\theta, \] we get \[ 2=\frac{\alpha}{2}-\frac{\sqrt3}{2}\beta, \] \[ 1=\frac{\sqrt3}{2}\alpha+\frac{\beta}{2}. \] Multiplying by \(2\), \[ 4=\alpha-\sqrt3\,\beta, \] \[ 2=\sqrt3\,\alpha+\beta. \]
Step 3: Solve for \(\alpha\) and \(\beta\). From \[ 4=\alpha-\sqrt3\,\beta, \] \[ \alpha=4+\sqrt3\,\beta. \] Substituting into the second equation, \[ 2=\sqrt3(4+\sqrt3\,\beta)+\beta. \] \[ 2=4\sqrt3+4\beta. \] \[ \beta=\frac{1-2\sqrt3}{2}. \] Therefore, \[ \alpha = 4+\sqrt3\left(\frac{1-2\sqrt3}{2}\right) = 1+\frac{\sqrt3}{2}. \]
Step 4: Find \(4\alpha+2\beta\). \[ 4\alpha+2\beta = 4\left(1+\frac{\sqrt3}{2}\right) + 2\left(\frac{1-2\sqrt3}{2}\right). \] \[ = 4+2\sqrt3+1-2\sqrt3. \] \[ =5. \]
Step 5: Write the final answer. \[ \boxed{5} \]
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