Find the numbers

Question

A positive real number is  more than another. If the sum of the squares of the two numbers is , find the numbers.

Answer 1

x=y+6, x^2 +y^2 = 54, y^2 +(y+6)^2 = 54

solve by substitution elimination or graph the line and circle and find the intersection points

2y^2 +12y +36 = 54

y^2 +6y +18 = 27

y^2 +6y = 9

y^2 +6y +9 = 9+9

(y+3)^2 = 18

y+3 = +/-3sqr2

y = 3sqr2 -3 or -3sqr2-3

y = about 3(.414) or -3(2.414)

y = about 1.242 or - 7.242

x = 3sqr2 +3 or -3sqr2+3

= about 7.242 or -1.242

(x,y) = (3sqr2+3, 3sqr2-3), or (-3sqr2+3, -3sqr2-3)= the two intersection points of the line & circle