Let generalised equation of a conic be: x^2 + y^2 + 2hxy + 2gx + 2fy + c = 0, derive a relation between h^2 and ab

To derive a relation between \( h^2 \) and the semi-major axis (\(a\)) and semi-minor axis (\(b\)) of the general conic equation:

\[
x^2 + y^2 + 2hxy + 2gx + 2fy + c = 0
\]

First, let's simplify and analyze the equation step-by-step, assuming it's the equation of a conic centered at the origin. The terms \(2gx\), \(2fy\), and \(c\) can be neglected for now since we're focusing on the geometric properties of the conic (ellipse or hyperbola), which depend on the shape, not the location. This leaves us with:

\[
x^2 + y^2 + 2hxy = 0.
\]

### Step 1: Transform into canonical form

To simplify further, we can eliminate the cross term \(2hxy\) by rotating the coordinate system. The general equation of a conic with a mixed term \(2hxy\) represents a conic that is rotated with respect to the \(x\)- and \(y\)-axes.

We apply a rotation of axes transformation:
\[
x' = x \cos \theta + y \sin \theta, \quad y' = -x \

Let the generalised equation of a conic be: x^2 + y^2 + 2hxy + 2gx + 2fy + c = 0, derive a relation between h^2 and ab.

Understand how to derive the relation between h² and the semi-major and semi-minor axes (a, b) in the generalized conic equation x^2 + y^2 + 2hxy + 2gx + 2fy + c = 0. This problem covers key concepts in conic sections and coordinate geometry, including rotation of axes.

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