College Trigonometry, 2011a

988 Applications of Trigonometry
5. 13x 2 − 34xy √ 3+47y 2 − 64 = 0
becomes (y ′ ) 2 − (x′ ) 2
16
= 1 after rotating
counter-clockwise through θ = π 6 .
y ′ θ = π 6
y
6. x 2 − 2 √ 3xy − y 2 +8=0
becomes (x′ ) 2
4
− (y′ ) 2
4
= 1 after rotating
counter-clockwise through θ = π 3
y x ′
x ′
y ′ θ = π 3
x
x
13x 2 − 34xy √ 3+47y 2 − 64 = 0
7. x 2 − 4xy +4y 2 − 2x √ 5 − y √ 5=0
becomes (y ′ ) 2 = x after rotating
counter-clockwise through θ =arctan ( 1
2)
.
y
x 2 − 2 √ 3xy − y 2 +8=0
8. 8x 2 +12xy +17y 2 − 20 = 0
becomes (x ′ ) 2 + (y′ ) 2
4
= 1 after rotating
counter-clockwise through θ =arctan(2)
y
x ′
y ′ x ′
θ = arctan ( ) 1
2
y ′
θ = arctan(2)
x
x
x 2 − 4xy +4y 2 − 2x √ 5 − y √ 5=0
8x 2 +12xy +17y 2 − 20 = 0