In Class 11, conic sections are studied as part of coordinate geometry, focusing on the properties and equations of curves formed by the intersection of a plane and a cone. The main conic sections studied are the circle, ellipse, parabola, and hyperbola. Here’s an overview:
1. **Circle**: A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is \((x – h)^2 + (y – k)^2 = r^2\).
2. **Ellipse**: An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. The standard form of the equation of an ellipse centered at the origin with major axis along the x-axis is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.
3. **Parabola**: A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard form of the equation of a parabola with vertex at the origin and axis parallel to the y-axis is \(y^2 = 4ax\) or \(x^2 = 4ay\), depending on the orientation.
4. **Hyperbola**: A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points (the foci) is constant. The standard form of the equation of a hyperbola centered at the origin with transverse axis along the x-axis is \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the distances from the center to the vertices and to the foci, respectively.
5. **Eccentricity**: The eccentricity of a conic section is a measure of how “open” or “closed” the curve is. It is defined as the ratio of the distance from a point on the conic section to a focus to the distance from that point to the corresponding directrix.
Studying conic sections helps in understanding the geometric properties of these curves and their applications in various fields, including physics, astronomy, and engineering.
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