Question

Use orthogonal transformations and translations to reduce the conic given by 5x² +12xy+10y²−6x+4y−1=0 to the canonical form.

Answer 1

To reduce the given conic 5x² +12xy+10y²−6x+4y−1=0 to its canonical form, we can use orthogonal transformations and translations.

Step-by-step explanation:

To reduce the given conic 5x² +12xy+10y²−6x+4y−1=0 to its canonical form, we can use orthogonal transformations and translations.

First, we need to eliminate the xy term by setting up a rotation matrix:

R = cos(θ) -sin(θ)

sin(θ) cos(θ)

By choosing the appropriate value of θ, we can cancel out the xy term and reduce the conic to its canonical form.

Next, we can use translations to eliminate the linear terms and reduce the equation to the canonical form.