Where the tangent is perpendicular to...?

Question

Find the points on the curve y = 2 - 1/X where the tangent is perpendicular to the line y = 1 - 4x.

Answer 1

The slope of y=1-4x is -4, so the slope of its perpendicular is 1/4. Thus, we must find a point where dy/dx = 1/4. Taking the derivative:

dy/dx = 1/x²
Setting this equal to 1/4:
1/x²=1/4
x²=4
x=±2

Answer 2

y = 2 - 1/x

dy/dx = 1/x²

Slope of required tangent = 1/4 (as 1/4 * -4 = -1)

so 1/x² = 1/4
x = ±2

When x = 2 y = 1.5
When x = -2 y = 2.5

So the two points are (2, 1.5) and (-2, 2.5)

Answer 3

slope, m, of straight line
= -4
The perpendicular to the line
= -1/m
= 1/4
=0.25

For the curve,
y=f(x)=2-1/x
dy/dx=f'(x)=1/x^2
We solve for
f'(x)=0.25
1/x^2=0.25
x=+/- 2

y(2)=2-1/2=1.5
y(-2)=2-1/(-2)=2.5
the points on the curve are:
(-2,2,5), (2,1.5)