If the axes are rotated through an angle 45∘ in the positive direction without changing the origin, then the co-ordinates of the point (√2,4) in the old system are
If θ is the angle of rotation, then the co-ordinates in the new system are x′=x‌cos‌θ+ysin‌θ, y′=y‌cos‌θ−xsin‌θ Given that x′=√2,y′=4 Thus, x‌cos‌θ+ysin‌θ=√2 y‌cos‌θ−xsin‌θ=4 Also, θ=‌
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4
⇒x‌cos‌
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4
+ysin‌‌
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4
=√2 and y‌cos‌
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4
−xsin‌‌
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4
=4 ⇒‌‌x+y=2 . . . (i) and ‌‌y−x=4√2 . . . (ii) On adding Eqs. (i) and (ii), we get 2y=2+4√2⇒‌‌y=1+2√2 On subtracting (i) and (ii), we get ‌2x=2−4√2 ‌⇒‌‌x=1−2√2 Thus, the co-ordinates of (√2,4) in the old system (1−2√2,1+2√2)