NOTE: My explanation is both much longer and more rigorous than the explanation given by the other tutor using symmetry alone, since my explanation uses extensive calculus and algebra to find the same answer:
The equation of the circle x2 + y2 = 9 means that y = +- √(9 - x2), and the distance formula between our 2 points (x, y) on the circle and (4, 5) is d(x, y) = √[ (x - 4)2 + (y - 5)2 ] = √[ (x - 4)2 + (+- √(9 - x2) - 5)2 ], so:
d(x) = √[ x2 - 8x + 16 + (9 - x2) +- 10 √(9 - x2) + 25 ] = √[ - 8x +- 10 √(9 - x2) + 50 ], so now we must
take the derivative and set it equal to zero to find the extreme points of d(x), namely maxima and/or
minima: d'(x) = 0.5 * [ - 8 +- 10 * 0.5 * (-2x) / √(9 - x2) ] / √[ - 8x +- 10 √(9 - x2) + 50 ] = 0, so:
- 8 +- 10x / √(9 - x2) = 0, since the rational function 0.5 / √[ - 8x +- 10 √(9 - x2) + 50 ] is never
equal to zero, since the distance, which is the denominator, is always finite, so +-10x / √(9 - x2) = 8,
so 1/√(9 - x2) = +-8/(10x), so √(9 - x2) = +-(10x)/8 = +-5x/4, so 9 - x2 = (+-5x/4)2 = 25x2/16, so
9 = (+-5x/4)2 = 25x2/16 + 16x2/16 = 41x2/16, so 9*16/41 = x2, so x = +-4*3/√(41) = +-12/√(41).
Now. we must find the distance for each of these x values and take the smallest of those:
d(+-12/√(41)) = √[ - 8 * {+-12/√(41)} +- 10 √(9 - {+-12/√(41)}2) + 50 ]
d(+-12/√(41)) = √[ +-96 / √(41) +- 10 √(9 - 144/41) + 50 ] = √[ 50 +-96 / √(41) +- 10 √(369/41 - 144/41) ]
d(+-12/√(41)) = √[ 50 +- 96 / √(41) +- 10 √(225/41) ] = √[ 50 +- 96 / √(41) +- 10*15 / √(41) ]
d(+-12/√(41)) = √[ 50 +- 96 / √(41) +- 150 / √(41) ], the lowest of which will be the following:
d(+-12/√(41)) [lowest] = √[ 50 - 96 / √(41) - 150 / √(41) ] = √[ 50 - 246 / √(41) ] = √[ 50 - 246√(41)/41 ].
Shortest distance = √[ 50 - 6√(41) ] = √[ 41 - 6√(41) + 9 ] = √[ {√(41)}2 - 2*3*√(41) + 32 ],
and, finally, we have that the shortest distance = √[ {√(41) - 3}2 ] = √(41) - 3.
