Question

Помогите пожалуйста решить задачку.

Чему равна площадь восьмиугольника, вершины которого являются решениями системы уравнений:
|x| + |y| = 7
x^2 + y^2 = 25

Answer 1

$left { {{|x|+|y|=7,} atop {x^2+y^2=25,}} right. \ (|x|+|y|)^2=x^2+2|x||y|+y^2, \ x^2+y^2=(|x|+|y|)^2-2|x||y|=7^2-2|xy|, \ left { {{|x|+|y|=7,} atop {49-2|x||y|=25,}} right. left { {{|x|+|y|=7,} atop {|x||y|=12,}} right. left { {{|x|+frac{12}{|x|}=7,} atop {|y|=frac{12}{|x|} ,}} right. , \ x^2-7|x|+12=0, \ left [ {{|x|=3,} atop {|x|=4,}} right. left[begin{array}{c}x=-3,\x=3,\x=-4,\x=4,end{array}right.$

$left[begin{array}{c} left { {x=-3,} atop {|y|=4,}} right. \ left { {x=3,} atop {|y|=4,}} right. \ left { {x=-4,} atop {|y|=3,}} right. \ left { {x=4,} atop {|y|=3,}} right. end{array}right.$

$left[begin{array}{c} left { {x=-3,} atop {y=-4,}} right. \ left { {x=-3,} atop {y=4,}} right. \ left { {x=3,} atop {y=-4,}} right. \ left { {x=3,} atop {y=4,}} right.\ left { {x=-4,} atop {y=-3,}} right. \ left { {x=-4,} atop {y=3,}} right. \ left { {x=4,} atop {y=-3,}} right. \ left { {x=4,} atop {y=3,}} right. end{array}right.$

$S=S_square-4S_triangle=8cdot8-4cdotfrac{1}{2}cdot1cdot1=64-2=62$