Given that x²+y²=14x+6y+6,find the largest possible value of the expression E=3x+4y?

x^2-14x +y^2-6y=6
(x-7)^2 +(y-3)^2 = 6+49+9 = 64 = 8^2
this is a circle centre (7,3) radius 8
y = 3 + 8 sin[theta]
x= 7+ 8 cos[theta]
E= 3(7+8cos( theta)) +4 (3 +8 sin[theta])
= 21 +24 cos (theta) + 12 +32 sin (theta)
= 33 +24 cos (theta)+ 32 sin(theta)
dE/ d (theta)= -24sin(theta) +32cos (theta)=0
32cos theta=24sin (theta)
tan (theta) = 32/24 = 4/3
theta = arctan(4/3)
E= 33 +24 cos (arctan(4/3)) + 32 sin(arctan(4/3)
=73

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