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# The origin of the co-ordinate axes is shifted to (-1,3) and the axes is rotated through an angle of 90∘ in anti-clockwise direction. If (a,b) is the new coordinates of (2,3) in the new coordinate system, then find the value of 2a2+3b2 __

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## We know when origin gets shifted to (h,k) the new coordinates of (x,y) will be (x-h,y-k). We also know when we rotate the coordinate axes through one angle of θ,(x+iy) becomes (x+iy)e−iθ, because it is equivalent to rotating (x+iy) through −θ. In this question we have to do both. Which one will you do first? Rotation or translation? Can we do it in any order we want ? We will try both 1)Rotation then translation. Our point is (2+3i) we will rotate it through −90∘ (2 + 3i) becomes (2+3i) e−iπ2=(2+3i)×−i = -2i + 3 = 3 - 2i ⇒ (3,-2) We will do the translation now, New origin is (-2,1) ⇒ (3,-2) becomes (3 - (-2),-2-1) = (5,-3) 2) Translation then rotation New origin is (-2,1) ⇒ (2,3) becomes (2 - (-2),3-1) = (4,2) Co-ordinate axes is rotated through 90∘ ⇒ we will rotate (4,2) or (4+2i) through −90∘ (4+2i)e−iπ2=(4+2i)−i. = -4i + 2 = 2 - 4i ⇒ (2,-4) When we did rotation first we got (5,-3) and when we did translation first we got (2,-4).Which one is correct (2,-4) is the correct one what was the mistake we did in the first method?  Suggest Corrections  0      Similar questions
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