Question

What does the equation x square -4xy+y square =0 become when the the axes are turned through an angle 45°

​

Answer 1

Step-by-step explanation:

$\color{red}{Answer}$

The relation between coordinates

(

x

,

y

)

and

(

x

'

.

y

'

)

can be expressed as

x

=

x

'

cos

θ

−

y

'

sin

θ

and

y

=

x

'

sin

θ

+

y

'

cos

θ

or

x

'

=

x

cos

θ

+

y

sin

θ

and

y

=

−

x

sin

θ

+

y

cos

θ

for this we need to have

θ

given by

cot

2

θ

=

A

−

C

B

In the given case as equation is

x

2

−

4

x

y

+

y

2

+

1

=

0

, we have

A

=

C

=

1

and

B

=

−

4

and hence

cot

2

θ

=

0

i.e.

θ

=

π

4

Hence relation is give by

x

=

x

'

cos

(

π

4

)

−

y

'

sin

(

π

4

)

and

y

=

x

'

sin

(

π

4

)

+

y

'

cos

(

π

4

)

i.e.

x

=

x

'

√

2

−

y

'

√

2

and

y

=

x

'

√

2

+

y

'

√

2

Hence, we get

(

x

'

√

2

−

y

'

√

2

)

2

−

4

(

x

'

√

2

−

y

'

√

2

)

(

x

'

√

2

+

y

'

√

2

)

+

(

x

'

√

2

+

y

'

√

2

)

2

+

1

=

0

or

(

x

'

2

2

+

y

'

2

2

−

x

'

y

'

)

−

4

(

x

'

2

2

−

y

'

2

2

)

+

(

x

'

2

2

+

y

'

2

2

+

x

'

y

'

)

+

1

=

0

or

−

x

'

2

+

3

y

'

2

+

1

=

0

or

x

'

2

−

3

y

'

2

=

1

The two graphs are as follows:

graph{x^2-4xy+y^2+1=0 [-10, 10, -5, 5]}

and

graph{x^2-3y^2=1 [-10, 10, -5, 5]}

Regards..

$\fcolorbox{blue}{azure} {Itzintellectual}$