# Point A is at #(-2 ,8 )# and point B is at #(-1 ,3 )#. Point A is rotated #(3pi)/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

The change in distance is:

The first distance from B to A is:

Compute the polar coordinates of point A:

Compute the x and y coordinates for new point A:

Compute the new distance:

The change in distance is:

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The new coordinates of point A after rotating ((3\pi)/2) clockwise about the origin are ((8, 2)).

To find the new coordinates, we use the following rotation formulas:

[ x' = x \cos(\theta) - y \sin(\theta) ] [ y' = x \sin(\theta) + y \cos(\theta) ]

For point A ((-2, 8)), substituting (\theta = (3\pi)/2), we get:

[ x' = -2 \cos\left(\frac{3\pi}{2}\right) - 8 \sin\left(\frac{3\pi}{2}\right) = 8 ] [ y' = -2 \sin\left(\frac{3\pi}{2}\right) + 8 \cos\left(\frac{3\pi}{2}\right) = 2 ]

So, the new coordinates of point A are ((8, 2)).

The original distance between points A and B can be calculated using the distance formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For points A ((-2, 8)) and B ((-1, 3)), the original distance is:

[ d_{\text{original}} = \sqrt{(-1 - (-2))^2 + (3 - 8)^2} = \sqrt{1^2 + 5^2} = \sqrt{26} ]

After the rotation, the new coordinates of point A are ((8, 2)). So, the distance between the new point A and B ((-1, 3)) is:

[ d_{\text{new}} = \sqrt{(8 - (-1))^2 + (2 - 3)^2} = \sqrt{9^2 + (-1)^2} = \sqrt{82} ]

The change in distance is given by:

[ \Delta d = d_{\text{new}} - d_{\text{original}} = \sqrt{82} - \sqrt{26} ]

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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.

- Point A is at #(-2 ,-4 )# and point B is at #(-3 ,3 )#. Point A is rotated #(3pi)/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?
- Circle A has a radius of #4 # and a center of #(5 ,3 )#. Circle B has a radius of #2 # and a center of #(1 ,2 )#. If circle B is translated by #<2 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- Point A is at #(7 ,-1 )# and point B is at #(-8 ,-2 )#. Point A is rotated #pi/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?
- Points A and B are at #(2 ,5 )# and #(6 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #1/2 #. If point A is now at point B, what are the coordinates of point C?
- 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?

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