Question

Point A\((5,12)\) rotated about the origin O in the XY-plane through an angle of \(30^{\circ}\) in the anticlockwise direction to a new position B. The ordinate of point B is...

• A. \(6\sqrt{3}+\frac{5}{2}\)
• B. \(\frac{5\sqrt{3}}{2}-6\)
• C. \(\frac{5\sqrt{3}}{2}+6\)
• D. \(6\sqrt{3}-\frac{5}{2}\)

Hint:

Rotating (x, y) by an angle t gives y-coordinate x sin t + y cos t.

Answer 1

The Correct Option is A

Step 1: Understand the concept
When a point \((x, y)\) is rotated about the origin anticlockwise by an angle \(\theta\), its new coordinates are \((x\cos\theta - y\sin\theta,\ x\sin\theta + y\cos\theta)\).

Step 2: Substitute the data
Here \(x = 5\), \(y = 12\) and \(\theta = 30^\circ\), with \(\sin 30^\circ = \frac{1}{2}\) and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).

Step 3: Compute the ordinate
\[ y_B = 5 \sin 30^\circ + 12 \cos 30^\circ = \frac{5}{2} + 6\sqrt{3} \]

Step 4: Check the other options
The abscissa is \(5\cos30^\circ - 12\sin30^\circ = \frac{5\sqrt{3}}{2} - 6\), which is option (B), so that option gives the abscissa and not the ordinate. Options (C) and (D) mix the signs. The ordinate is \(6\sqrt{3} + \frac{5}{2}\).

Final Answer:
The ordinate of B is 6 sqrt(3) + 5/2. This is option (A). \[ \boxed{\text{(A) }6\sqrt{3}+\frac{5}{2}} \]