Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term. 10x2 - 4xy + 6y2 - 8x + 8y = 0

x = x' cos  (π/8) + y' sin  (π/8)

y = -x' sin (π/8) + y' cos (π/8)

Step-by-step explanation:

Canonical form of conics section:

A*x^2 + B*x + C*y^2 + D*y + E*x*y + F = 0

We want that in the rotated system the equation has no x'y'-term. To do this the rotated angle has to satisfy:

tan(2 θ) = E/(C - A)

The rotation formula when the coordinates system rotates an angle  θ  are:

x = x' cos  θ + y' sin  θ

y = -x' sin θ + y' cos θ

The conic section is: 10*x^2 - 4*x*y + 6*y^2 - 8*x + 8*y = 0, then:

A = 10

B = -8

C = 6

D = 8

E = -4

F = 0

So,

tan(2 θ) = -4/(6 - 10)

2 θ = tan^-1 (1)

θ = (π/4)*(1/2) = π/8

Finally,

x = x' cos  (π/8) + y' sin  (π/8)

y = -x' sin (π/8) + y' cos (π/8)

he can buy 4lbs. of live bait and 1lbs of natural bait

step-by-step explanation:

answer: (yeees, you need a poet.)

//blooming serenity//

beautiful in the morning

mysterious at night

wonderous nature souring

with all it's mystical might

the flower that blooms

to the fox that looms

nocturnal creatures

with various features

a blooming flower soon in sight

as a new little hatchling fears to take flight  

afraid to leave it's calming nest,

but dear little bird, mother knows best

as sure as there are stars in the sky,

life is not about who, where what, when or why-

it's the journey that follows

adventures we swallow

of   nature's deep wonders,

natural disasters and storm's thunders.

blooming serenity

life and nature, truly is heavenly.

-kerstien lehman

step-by-step explanation: