a-b-x-y-R-a-b-23-ax-by-79-ax-2-by-2-217-ax-3-by-3-661-Find-ax-4-by-4-

Question Number 204129 by hardmath last updated on 06/Feb/24

a, b, x, y \in R

a + b = 23

ax + by = 79

{ax}^{2} + {by}^{2} = 217

{ax}^{3} + {by}^{3} = 661

Find : {ax}^{4} + {by}^{4} = ?

Answered by AST last updated on 06/Feb/24

x (a x + b y) = 79 x \Rightarrow {a x}^{2} + b x y = 79 x

a x y + {b y}^{2} = 79 y \Rightarrow {a x}^{2} + {b y}^{2} + x y (a + b) = 79 (x + y)

\Rightarrow 217 = 79 (x + y) - 23 x y \dots (i)

{a x}^{3} + {b x y}^{2} = 217 x; {a x}^{2} y + {b y}^{3} = 217 y

661 + x y (b y + a x) = 217 (x + y)

\Rightarrow 661 = 217 (x + y) - 79 x y \dots (i i)

{a x}^{4} + {b y}^{4} = x ({a x}^{3} + {b y}^{3}) + y ({a x}^{3} + {b y}^{3}) - x y ({b y}^{2} + {a x}^{2})

= 661 x + 661 y - 217 x y

23 (i i) - 79 (i) \Rightarrow - 1250 (x + y) = - 1940 \Rightarrow x + y = \frac{194}{125}

79 (i i) - 217 (i) \Rightarrow 5130 = - 1250 x y \Rightarrow x y = - \frac{513}{125}

\Rightarrow {a x}^{4} + {b y}^{4} = 661 \times \frac{194}{125} + \frac{217 \times 513}{125} = \frac{47911}{25}

Answered by mr W last updated on 06/Feb/24

(a x + b y) (x + y) = {a x}^{2} + {b y}^{2} + (a + b) x y

\Rightarrow 79 (x + y) = 217 + 23 x y \dots (i)

({a x}^{2} + {b y}^{2}) (x + y) = {a x}^{3} + {b y}^{3} + (a x + b y) x y

\Rightarrow 217 (x + y) = 661 + 79 x y \dots (i i)

23 \times (i i) - 79 \times (i) :

(23 \times 217 - 79 \times 79) (x + y) = 23 \times 661 - 79 \times 217

\Rightarrow x + y = \frac{194}{125}

\Rightarrow x y = \frac{1}{23} \times (79 \times \frac{194}{125} - 217) = - \frac{513}{125}

({a x}^{3} + {b y}^{3}) (x + y) = {a x}^{4} + {b y}^{4} + ({a x}^{2} + {b y}^{2}) x y

\Rightarrow {a x}^{4} + {b y}^{4} = 661 \times \frac{194}{125} + 217 \times \frac{513}{125} = \frac{47911}{25}

Commented by Tawa11 last updated on 06/Feb/24

Nice sir

Commented by mr W last updated on 06/Feb/24

g e n e r a l l y

w i t h s_{n} = {a x}^{n} + {b y}^{n}

s_{n} = (x + y) s_{n - 1} - {x y s}_{n - 2}

s_{4} = \frac{194}{125} \times 661 + \frac{513}{125} \times 217 = \frac{47911}{25}

s_{5} = \frac{194}{125} \times \frac{47911}{25} + \frac{513}{125} \times 661 = \frac{17772059}{3125}

\dots \dots

Commented by hardmath last updated on 06/Feb/24

perfect solution dear prafessor thank you