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Question 1014504: The line 2x-y=5 turns about the point on it, whose ordinate and abscissae are equal, through an angle of 45°, in anti clockwise direction. Find the equation of line in the new position
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! not sure how you're supposed to solve this, but this is how i did it.
your original equation is 2x - y = 5
solve for y to get y = 2x - 5
since the slope intercept form of the equation of a straight line is y = mx + b, and the slope is m and the y-intercept is b, then your slope is 2.
that's also the tangent of the angle between your line and a horizontal line that goes through any point on your line.
so you have tangent (theta) = 2.
that makes the angle theta equal to 63.42494882 degrees.
now you are at the point (5,5).
draw a horizontal line through it and draw the line of your equation through it.
now, if you rotate that line an additional 45 degrees in an anti-clockwise direction, the angle between the horizontal line and your new line will be 65... + 45 = 108.4349494882 degrees.
that means that the tangent of your new line will be tangent (108.4349494882).
solve for that to get your new tangent equals -3.
that means the slope of your new line is equal to -3.
the slope intercept form of an equation for a straight line is y = mx + b
m is the slope and b is the y intercept.
when the slope is -3, the equation becomes y = -3x + b
you need to find b.
when y = -3x + b goes through the point (5,5), you can replace y with 5 and x with 5 to get:
5 = -3*5 + b
solve for b to get b = 20
the equation of your new line is y = -3x + 20.
it will intersect with your oirignal line at (5,5) and the angle between it and the original line will be 45 degrees.
the graph of your original line and the new line is shown below:
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