Chapter 6

इकाई 1 : X-Y में व्यापक द्विघाती कार्तीय समीकरण का उनके मानक रूप में समानयन (Reduction of general equation of second degree Cartesian equation in X-Y to their standard forms)

This section introduces the reduction of the general equation of the second degree in two variables (x and y) to its standard forms. The general equation of the second degree in x and y is given by: ax² + 2hxy + by² + 2gx + 2fy + c = 0 This equation represents various conic sections such as straight lines, points, parabolas, hyperbolas, and ellipses, depending on the values of the coefficients. The main objective of this unit is to determine the form of the conic section represented by a given general quadratic equation and to reduce it to its standard form. The section also covers how to find the center of the conic section and how to write its equation with respect to axes passing through the center. The outline of the unit includes definitions, transformation of axes by rotation, the general equation of conic sections, theorems, special cases, the center of the conic, and conditions for the equation to represent a conic section.

• The general quadratic equation in x and y represents conic sections.
• Reduction to standard form helps identify the type of conic section.
• The center of the conic can be found from the equation.
• Transformation of axes is used to simplify the equation.
• 📌 Conic Section: A curve obtained as the intersection of a plane with a double-napped cone.
• 📌 Standard Form: The simplified form of the equation representing a conic section.
• 📌 Center: The point that is equidistant from all points on the conic section (for ellipses and hyperbolas).

1.0 उद्देश्य (Objectives)

The objective of this unit is to enable students to reduce the general equation of the second degree in x and y, namely ax² + 2hxy + by² + 2gx + 2fy + c = 0, to the standard forms of conic sections such as straight lines, points, parabolas, and hyperbolas. Students will learn to determine the type of conic section represented by a given equation and to find the center of the conic. The unit also covers how to express the equation of the conic with respect to axes passing through the center.

• Learn to reduce the general quadratic equation to standard forms.
• Identify the type of conic section from the equation.
• Find the center of the conic section.
• Express the equation with respect to axes through the center.
• 📌 Standard Form: The canonical representation of a conic section.
• 📌 Center: The point about which the conic is symmetric.