Question

The origin of the co-ordinate axes is shifted to (-1,3) and the axes is rotated through an angle of ${90}^{\circ }$ in anti-clockwise direction. If (a,b) is the new coordinates of (2,3) in the new coordinate system, then find the value of $2a^2+3b^2$

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Answer 1

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 $\theta$,(x+iy) becomes (x+iy)$e^{-i\theta }$, because it is equivalent to rotating (x+iy) through $-\theta$.

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 $-{90}^{\circ }$

(2 + 3i) becomes (2+3i) $e^{\frac{-i\pi }{2}}=(2+3i)\times -i$

= -2i + 3

= 3 - 2i

$\Rightarrow$ (3,-2)

We will do the translation now,

New origin is (-2,1)

$\Rightarrow$ (3,-2) becomes (3 - (-2),-2-1) = (5,-3)

2) Translation then rotation

New origin is (-2,1)

$\Rightarrow$ (2,3) becomes (2 - (-2),3-1) = (4,2)

Co-ordinate axes is rotated through ${90}^{\circ }$

$\Rightarrow$ we will rotate (4,2) or (4+2i) through $-{90}^{\circ }$

$(4+2i)e^{\frac{-i\pi }{2}}=(4+2i)-i$.

= -4i + 2

= 2 - 4i

$\Rightarrow$ (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?