To determine the new coordinates of point
B after it has been rotated about point
A through an angle of
15∘ in the anticlockwise direction, we can use rotation formulas. First, let's denote the original coordinates as:
A(2,0)B(3,1)Let's transform point
B to the origin by translating point
A to the origin:
In vector notation, point
B relative to
A is:
=(3−2,1−0)=(1,1) After the rotation, the new coordinates
C can be determined using the rotation matrix. The rotation matrix for a counterclockwise rotation by angle
θ is:
(| cos‌θ | −sin‌θ |
| sin‌θ | cos‌θ |
) For an angle of
15∘, we have:
θ=15∘The cosine and sine of
15∘ are:
‌cos‌15∘=‌‌sin‌15∘=‌Using the rotation matrix:
(| cos‌15∘ | −sin‌15∘ |
| sin‌15∘ | cos‌15∘ |
)() This multiplication gives us:
(| ‌⋅1−‌⋅1 |
| ‌⋅1+‌⋅1 |
) This simplifies to:
Now, we revert the translation back by adding
A(2,0) to get the coordinates of
C :
C=(2+‌,√‌)The correct option is:
Option D
(2+‌,√‌)