Point A is at $(4 ,-2 )$ and point B is at $(1 ,-3 )$ . 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?

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Answer 1 · 2016-02-17T03:05:28

The new point $A(-2, -4)$
and No change in distance. Same distance $=sqrt10$

the equation passing thru the origin (0, 0) and point A(4, -2) is

$y=-1/2 x$

and distance from (0, 0) to A(4, -2) is $=2sqrt5$

the equation passing thru the origin (0, 0) and perpendicular to line $y=-1/2 x$ is

$y=2x" " "$ first equation

We are looking for a new point $A(x_1, y_1)$ on the line $y=2x$ at the 3rd quadrant and $2sqrt5$ away from the origin (0, 0).

Use distance formula for "distance from line to a point not on the line"

$d=(ax_1+by_1+c)/(+-sqrt(a^2+b^2)$

Use equation, $y=-1/2x$ which is also $x+2y=0$

$a=1$ and $b=2$ and $c=0$

$d=2sqrt5=(x_1+2y_1)/(+-sqrt(1^2+2^2))$

$2sqrt5=(x_1+2y_1)/(-sqrt(5))" " "$ choose negative because,
$(x_1, y_1)$ is below the line.

$-2sqrt5*sqrt5=x_1+2y_1$

$-10=x_1+2y_1" " "$ second equation

$y_1=2x_1" " "$ first equation

Using first and second equation. Solve for $(x_1, y_1)$

and $x_1=-2$ and $y_1=-4$

Point A is at (4 ,-2 )(4 ,-2 ) and point B is at (1 ,-3 )(1 ,-3 ). Point A is rotated pi/2 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?