Section III.1

§ 1

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Problem III.1.1

Reduce to canonical form the equation

3x2+10xy+3y2−2x+2y−9=0. 3 x^{2}+10 x y+3 y^{2}-2 x+2 y-9=0 .
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Problem III.1.2

Reduce to canonical form the equation

9x2−24xy+16y2−34x−38y−9=0. 9 x^{2}-24 x y+16 y^{2}-34 x-38 y-9=0 .
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Problem III.1.3

What class of curve can a second-order equation define if its left-hand side factors into a product of linear polynomials?

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Problem III.1.4

Under what necessary and sufficient condition on its coefficients is a second-order equation in a Cartesian rectangular coordinate system the equation of a circle?

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Problem III.1.5

A coordinate system satisfies the conditions ∣e1∣=∣e2∣=5,(e1,e2)=7\left|\mathbf{e}_{1}\right|=\left|\mathbf{e}_{2}\right|=5,\left(\mathbf{e}_{1}, \mathbf{e}_{2}\right)=7. What curve is defined in this coordinate system by the equation x2+y2=1x^{2}+y^{2}=1 ?

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Problem III.1.6

Prove that the sum of the coefficients A+CA+C in the general second-order equation does not change under the passage from one Cartesian rectangular coordinate system to another such system.

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