Question

Determine the equation of the parabola
x ^2= 2xy + y ^2+ 2x - 4y + 3 = 1
after rotating of axes is through
45°
​

Answer 1

Step-by-step explanation:

Let axes be rotated through an angle ϕ, then old co-ordinates are

x=x

1

cosϕ−y

1

sinϕ

y=x

1

sinϕ+y

1

cosϕ

∴(x

1

cosϕ−y

1

sinϕ)

2

+2

3

(x

1

cosϕ−y

1

sinϕ)(x

1

sinϕ+y

1

cosϕ)−(x

1

sinϕ+y

1

cosϕ)

2

=2a

2

coefficient of x

1

y

1

is

−2cosϕsinϕ+2

3

(cos

2

ϕ−sin

2

ϕ)−2cosϕsinϕ=0

2

3

(cos2ϕ)=2sin2ϕ

tan2ϕ=

3

⇒2ϕ=

3

π

ϕ=

6

π

∴ angle through which it is to be rotated is

6

π