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Question 709679: find the tangent line to the ellipse x^2+2y^2=9 perpendicular to the line 4x-y=6
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! find the tangent line to the ellipse x^2+2y^2=9 perpendicular to the line 4x-y=6
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The slope of the line m = 4
Lines perpendicular have a slope of -1/4
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Find the 2 points on the ellipse with a slope of -1/4
Find the slope of the tangent at any point on the ellipse:
2xdx + 4ydy = 0
dy/dx = -x/2y
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-x/2y = -1/4
y = 2x
The points are the solutions to the system
y = 2x
x^2 + 2y^2 = 9
Sub for y
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x^2 + 8x^2 = 9
x^2 = 1
x = +1
y = 2x = 2
--> (1,2)
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Also (-1,-2)
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Use y = mx + b to find b
2 = (-1/4)*1 + b
b = 9/4
--> y = -x/4 + 9/4 tangent to (1,2)
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-2 = (-1/4)*(-1) + b
b = -9/4
--> y = -x/4 - 9/4 tangent to (-1,-2)
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