Find the equation of the line perpendicular to the line tangent to y = 2x - x^2 at x = 1/2?

Question

I solved it but the answer choices are totally different
here are the answer choices for them.
a.4y - 3 = 0
b.4x + 4y - 5 = 0
c. 2x-2y -1 = 0
d. 4x - 4y +1 = 0
e. 2x + 2y - 1 = 0

the answer i got was D, but I dont think it's right because I didnt change it into perpendicular. Please help me.

Answer 1

y = 2x - x^2
dy/dx = 2 - 2x

When x = 1/2,

y = 2x - x^2 = 2(1/2) - (1/2)^2 = 3/4
dy/dx = 2 - 2(1/2) = 1

So you're finding the equation of the line which passes through (1/2, 3/4), and has a gradient of -1/1 = -1.

y - 3/4 = -1(x - 1/2)
y - 3/4 = -x + 1/2
4y - 3 = -4x + 2
4x + 4y - 5 = 0

So it's B.

Answer 2

dy/dx = 2 - 2x, so at x = 1/2 the slope of the tangent is 1, so the perpendicular at this point has slope -1. The line with slope -1 and passing through (1/2, 3/4) is
y - 3/4 = -1(x-1/2) or simply y = -x + 5/4, which, if we multiply through by 4 and write in standard form gives 4x+4y-5 = 0, choice (b).

Answer 3

The radius is the gap from (a million, 0) to the line 3x - 4y = 24. 3x - 4y - 24 = 0 d = |3(a million) - 4(0) - 24|/?(3^2 + (-4)^2) = |-21|/?(9 + sixteen) = 21/5 (x - a million)^2 + y^2 = 441/25 25x^2 - 50x + 25 + 25y^2 = 441 25x^2 + 25y^2 - 50x - 416 = 0

Answer 4

f `(x) = 2 - 2x
f `(1/2) = 2 - 1 = 1
m1 = 1 
m2 = - 1
f(1/2) = 1 - 1/4 = 3/4

y - 3/4 = (-1) (x - 1/2)
4y - 3 = - 4x + 2
4x + 4y - 5 = 0
ANSWER b