whats the equation of the two lines that are perpendicular to y=.25x and tangent to the curve xy=1.?

Question

i need a mathematical answer

Answer 1

First, look at where the curve intersects the line:

curve y=1/x
line y=x/4

1/x=x/4
x=+/- 2

Note that the curve is symmetrical about the origin for x&y positive and x&y negative, so the line intersects twice.

That's actually somewhat irrelevant, but it helped me visualize the line and the curve.

Take the first derivative of the curve
dy/dx=-1/x^2
This is the instantaneous slope of the curve at any point, which is the tangent by formal definition.

A line perpendicular to the line y=.25x will have a slope of -4.
So set the derivative equal to -4 and solve for x
-4=-1/x^2
x=+/- 1/2
solve for y
and find the points
(1/2,2)
(-1/2,-2)
This is where the two lines that are both tangent to the curve and perpendicular to the line intersect the line.

one line is y=-4x+4
the other is y=-4x -4

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