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When the axes are rotated through 30 in the anticlockwise sense without changing  the origin, find the new coordinates of the point(-2,4).

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Let (x, y) be the old coordinates and (X, Y) be the new coordinates. Therefore

x = X cosθ - Y sin θ ...(1)

and y = X sinθ + Y cos θ   ...(2)

Solving Eq. (1) , We get

-2 = X cos 30° - Y sin 30°

⇒ -2 = √3X - Y/2

⇒ √3X - y = -4

 Solving Eq. (2), we get

4 = X sin30° + Y cos 30°

⇒ 4 = X + √3Y/2

⇒ X + √3Y = 8  ...(4)

Solving Eqs. (3) and (4), we get X = 2 - √3 and Y = 1 + 2√3  are the new coordinates of (-2, 4).

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