Question 1081702
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.