Let the second-degree equation a x^2+2 h x y+b y^2+2 g x+2 f...

Let the second-degree equation $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represent an ellipse. Which of the following is are invariant under the shifting of origin and the rotation of co-ordinate axes? (i) $\mathrm{h}^{2}-\mathrm{ab}$ (ii) $a+b$ (iii) eccentricity (iv) coordinates of foci (a) (i) and (ii) (b) (iii) and (iv) (c) (i), (ii) and (iii) (d) all the four.

Let the second-degree equation $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represent an ellipse. Which of the following is are invariant under the shifting of origin and the rotation of co-ordinate axes?
(i) $\mathrm{h}^{2}-\mathrm{ab}$
(ii) $a+b$
(iii) eccentricity
(iv) coordinates of foci
(a) (i) and (ii)
(b) (iii) and (iv)
(c) (i), (ii) and (iii)
(d) all the four.

Key Concepts

Eccentricity

Eccentricity is a measure that quantifies the deviation of a conic section from being circular. It is a fundamental invariant because it describes the intrinsic shape of the conic (for instance, how 'stretched' an ellipse is) regardless of any translation or rotation of the coordinate axes.

Coordinates of Foci

The coordinates of the foci are specific points related to the conic's geometry, particularly for ellipses and hyperbolas. Unlike the other parameters, these coordinates depend on the particular placement of the coordinate system and thus are not invariant under translations or rotations. Their values change when the axes are moved or reoriented, even though the relative position of the foci within the conic remains the same.

Invariant under Coordinate Transformations

This concept refers to properties of a conic section that remain unchanged even when the coordinate system undergoes translations (shifting of the origin) or rotations (changing the orientation of the axes). It is central in analytic geometry for recognizing features that are intrinsic to the conic, rather than artifacts of its representation.

Quadratic Form Coefficients and Their Invariants

In the general second-degree equation representing a conic, the coefficients of the quadratic terms can be arranged in a symmetric matrix. The trace (a + b) and the determinant-like expression (h² - ab) are invariant under rotation because they correspond to the eigenvalues of the quadratic form. These invariants are useful for classifying the conic and understanding its structure irrespective of the coordinate system used.

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