A maths

1)
let (a, a^2) be the point nearest to (0,1)
distance d = sqrt[ a^2 + (a^2 - 1)^2 ] = sqrt(a^4 - a^2 + 1)
let D = d^2 = a^4 - a^2 + 1
dD/da = 4a^3 - 2a
d^2 D/da^2 = 12a^2 - 2

set dD/da = 0, 4a^3 - 2a = 0
a = 0, a = 1/sqrt(2), a = -1/sqrt(2)

(d^2 D/da^2 when a = 0) = -2 < 0
( d^2 D/da^2 when a = 1/sqrt(2) ) > 0
( d^2 D/da^2 when a = -1/sqrt(2) ) > 0

( 1/sqrt(2), 1/2 ), ( -1/sqrt(2), 1/2 ) are points nearest to (0,1)

2)
a)
(x + 6)(y + 8) = 1728
y = 1728/(x + 6) - 8

b)
area of printing A
= xy
= 1728x/(x + 6) - 8x
= (-8x^2 + 1680x) / (x + 6)

dA/dx = [ (-16x + 1680)(x + 6) - (-8x^2 + 1680x) ] / (x + 6)^2
= (-8x^2 - 96x + 10080) / (x + 6)^2

set dA/dx = 0, (-8x^2 - 96x + 10080) = 0
x = 30 or x = -42 (rejected)

x = 30 should make A max.
you can check it yourself

dimensions = 36(wide) x 48(high)