Find all pairs of real numbers (a, b) such that (x-a)^2 + (2x-b)^2 = (x-3)^2 + (2x)^2 for all x.
+118702 (x−a)2+(2x−b)2=(x−3)2+(2x)2
I'd start by expanding it out. Have you done that? Show us.
+118702 Nice work guest !
x2+a2−2ax+4x2+b2−4bx=x2+9−6x+4x2
Now simply the left side and simplify the right side independently of each other.
x2+a2−2ax+4x2+b2−4bx=x2+9−6x+4x25x2+(−2a−4b)x+(a2+b2)=5x2−6x+9
Now equate coefficients
5=5 −2a−4b=−6a+2b=3a=3−2b (3−2b)2+b2=99+4b2−12b+b2=95b2−12b=0b(5b−12)=0b=0orb=12/5=2.4 ifb=0,a=3−0=3ifb=2.4,a=3−2∗2.4=3−4.8=−1.8so I have (0,3)and(2.4,−1.8)
You need to check these by plugging them into the ORIGINAL equation.
