Points A and B are at #(4 ,1 )# and #(3 ,9 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #4 #. If point A is now at point B, what are the coordinates of point C?

Answer 1

#C=(1/3,-25/3)#

#"under a counterclockwise rotation about the origin of "(3pi)/2#
#• " a point "(x,y)to(y,-x)#
#rArrA(4,1)toA'(1,-4)" where A' is the image of A"#
#rArrvec(CB)=color(red)(4)vec(CA')#
#rArrulb-ulc=4(ula'-ulc)#
#rArrulb-ulc=4ula'-4ulc#
#rArr3ulc=4ula'-ulb#
#color(white)(rArr3ulc)=4((1),(-4))-((3),(9))#
#color(white)(rArr3ulc)=((4),(-16))-((3),(9))=((1),(-25))#
#rArrulc=1/3((1),(-25))=((1/3),(-25/3))#
#rArrC=(1/3,-25/3)#
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Answer 2

To find the coordinates of point C, we first need to perform the rotation and dilation operations on point A.

  1. Rotation Counterclockwise by ( \frac{3\pi}{2} ): To rotate point A counterclockwise by ( \frac{3\pi}{2} ) about the origin, we use the following rotation matrix: [ \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} ] Substituting ( \theta = \frac{3\pi}{2} ), we have: [ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} ] Multiplying this matrix by the coordinates of point A, we get the coordinates of the rotated point.

  2. Dilation about Point C by a Factor of 4: After obtaining the coordinates of the rotated point, we dilate it by a factor of 4 about point C. This involves multiplying the coordinates of the rotated point by 4 and adding the coordinates of point C.

By performing these operations, we'll obtain the coordinates of point C.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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