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Question

If the axes are rotated through an angle of $$30^o$$ in the anti-clockwise direction, the coordinates of point $$(4, -2\sqrt 3)$$ with respect to new axes are


A
(2,3)
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B
(3,5)
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C
(2,3)
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D
(3,2)
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Solution

The correct option is B $$(\sqrt 3, -5)$$
The relation between coordinates $$(x,y) and (x'.y')$$ can be expressed as
$$x'=x\cos\theta+y\sin\theta$$
$$y'=-x\sin\theta+y\cos\theta$$

$$x'=(4)\cos\,30^o+(-2\sqrt{3})\sin 30^o$$
    $$=4\times\dfrac{\sqrt{3}}{2}-2\sqrt{3}\times\dfrac{1}{2}$$
    $$=2\sqrt{3}-\sqrt{3}$$
    $$=\sqrt{3}$$

$$y'=-(4)\sin(30^o)+(-2\sqrt{3})\cos(30^o)$$
$$=-4\times \dfrac{1}{2}-2\sqrt{3}\times\dfrac{\sqrt{3}}{2}$$    $$=-2+(-3)$$
$$=-5$$

$$\therefore$$  $$(x',y')=(\sqrt{3},-5)$$
$$\therefore$$  The required co-ordinates are $$(\sqrt{3},-5)$$

1481633_258194_ans_496815f984fd4da296ee830f8b5ad7b6.png

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