Question

Points A and B are at `(6 ,7 )` and `(3 ,9 )`, respectively. Point A is rotated counterclockwise about the origin by `(3pi)/2` and dilated about point C by a factor of `5`. If point A is now at point B, what are the coordinates of point C?

Answer 1

we know , if in two dimension the rotation of a point (x,y) about origin by an angle `theta` in anticlockwise direction transforms its coordinates into (x',y') then

`x'=xcostheta-ysintheta`

`y'=xsintheta+ycostheta`

Here `theta=(3pi)/2`
`costheta=cos((3pi)/2)=0 and sintheta=sin((3pi)/2)=-1`

So transformed coordinates of `A->(6,7)` will be

`A'->((6*0-7*(-1)),(6*(-1))+7*0))`

`=(7,-6)`

Similarly transformed coordinates of `B->(3,9)` will be

`B'->((3*0-9*(-1)),(3*(-1))+9*0))`

`=(9,-3)`
Let the coordinates of center of dilation C be (h,k).

So A' on 5 times dilation about C will be transformed into

`A'_"5xdilated"=(5(7-h)+h,5(-6-k)+k)`

By the given condition `A'_"5xdilated"=B`
So

`5(7-h)+h=3=>4h=32=>h=8`

Again

`5(-6-k)+k=9=>4k=-39=>k=-39/4`

Hence coordinates of `C->(8,-39/4)`