Let generalized equation of a conic be: ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0, derive a relation between h^2 and ab

The general equation of a conic is given by:

\[
ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0
\]

To derive a relation between \( h^2 \) and \( ab \), we analyze the discriminant of this equation, which determines the nature of the conic. The discriminant \( \Delta \) for a general second-degree equation of the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) (where \( A = a \), \( B = 2h \), \( C = b \)) is given by:

\[
\Delta = B^2 - 4AC
\]

Substituting \( A = a \), \( B = 2h \), and \( C = b \), we get:

\[
\Delta = (2h)^2 - 4ab = 4h^2 - 4ab
\]

Simplifying:

\[
\Delta = 4(h^2 - ab)
\]

Thus, for the discriminant, we have the following relation:

\[
\Delta = 4(h^2 - ab)
\]

This relation shows the connection between \( h^2 \) and \( ab \). The discriminant determines the type of conic:
- If \( \Delta > 0 \), the conic represents a hyperbola.
- If \( \Delta = 0 \), the conic represents a parabola.
- If \( \Delta < 0 \), the conic represents an ellipse (or a circle if \( a = b \) and \( h = 0 \)).

Hence, the derived relation between \( h^2 \) and \( ab \) is:

\[
h^2 - ab = \frac{\Delta}{4}
\]

This is the key relation governing the nature of the conic.

Would you like further details on the implications of this relation?

Here are 5 related questions:
1. How does the discriminant classify different types of conic sections?
2. What is the physical meaning of parameters \( a \), \( b \), and \( h \) in the conic equation?
3. Can we derive the equation of a rotated conic using this general form?
4. How does the condition \( h^2 = ab \) relate to the conic being a parabola?
5. What role does the discriminant play in determining the axes of the conic?

**Tip**: Always check the discriminant to determine the nature of a conic section when working with its general equation.

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

Learn how to derive the relation between h^2 and ab from the generalized equation of a conic, ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0. This solution covers key concepts in analytic geometry, including the discriminant and the classification of conic sections.

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