Question

已知点P(x,y)是圆x^2+y^2-10x+9=0上的点,求x^1+4x+y^2-2y最大值

Answer 1

x^2+y^2-10x+9=0

x^2-10x+25+y^2=16

(x-5)^2+y^2=16

令x=5+4cosa

y=4sina

x^2+4x+y^2-2y

=(5+4cosa)^2+4(5+4cosa)+16sin^2a-8sina

=61+56cosa-8sina

=61+8(7cosa-sina)

=61+8√(49+1)cos(a+φ).其中tanφ=1/7,不影响结果

=61+200√2cos(a+φ)

最大值=61+200√2