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Question 1081702: Find the equation that passes through (7,-4) at a distance of 1 unit from point (2,1).
Found 2 solutions by addingup, Alan3354:
Answer by addingup(3677)   (Show Source):
You can put this solution on YOUR website! One point: x, y
Another : x, y
At a distance of 1 unit in which direction, x or y? Is it (1, 1)or (2, 0) or something else?
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Here's your equation, in detail, for (7, 4)(2, 1). With this example you should be able to solve it once you figure out the one-unit issue:
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Slope of the line passing through two points is given by:
m = (y_2 - y_1)/(x_2 - x_1)
And you have:
x_1 = 7, y_1 = 4 and x_2 = 2, y_2 = 1 plug these values into the formula:
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m = ((1) - (4))/((2) - (7)) = -3/-5 = 3/5 (because -/- = +)
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The y-intercept is b = y_1 - m × x_1 Using your numbers:
b = 4 - (3/5) × 7 = -1/5
Now we are ready to write the equation in the form y = mx+b:
y = 3/5x - 1/5
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! Find the equation that passes through A(7,-4) at a distance of 1 unit from point B(2,1).
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There are 2 lines.
They pass thru point A, and are tangent to a circle of radius 1 centered at (2,1).
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The angles at the tangent point between the line and radii are 90 degs.
2 right triangles are formed.
The hypotenuse, c, is the distance from A to B.
c = 5sqrt(2)
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The short side of the right triangles = 1.
The long sides, from A to the tangent points, = 7 units.
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Find the 2 tangent points:
The 2 points are the intersection of the 1 unit circle and a 7 unit circle centered at A.
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The unit circle is (x-2)^2 + (y-1)^2 = 1
The other circle is (x-7)^2 + (y+4)^2 = 49
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Solving those 2 will be messy.
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Do it like this:
Find the 2 angles at A:
tan = 1/7
The slope of AB is -1 --> the angle between the x-axis and AB = 135 degs.
The angles between the 2 lines and the x-axis are 135 ± atan(1/7)
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tan(135 + atan(1/7)) = (-1 + 1/7)/(1 - (-1)*(1/7)) = (-6/7)/(8/7) = -3/4
The tangent is the slope.
slope = -3/4
y+4 = (-3/4)*(x-7) is one of the lines.
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tan(135 - atan(1/7)) = (-1 - 1/7)/(1 + (-1)*(1/7)) = (-8/7)/(6/7) = -4/3
y+4 = (-4/3)*(x-7) is the other line.
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