The conic equation of an ellipse is x 2/a 2 + y 2/b 2 = 1, and the equation of the auxiliary circle is x 2 + y 2 = a 2. Auxilary Circle: A circle drawn on the major axis of the ellipse as its diameter is called the auxiliary circle.Pole and Polar: For a point which is referred as a pole and lying outside the conic section, the locus of the points of intersection of the tangents, draw at the ends of the chords, drawn from this point is called the polar. We can have one normal for each of the tangents to the conic.Ĭhord of Contact: The chord drawn to join the point of contact of the tangents, drawn from an external point to the conic is called the chord of contact. Normal: The line drawn perpendicular to the tangent and passing through the point of contact and the focus of the conic is called the normal. Also from an external point, about two tangents can be drawn to the conic. The point where the tangent touches the conic is called the point of contact. Tangent: The tangent is a line touching the conic externally at one point on the conic. And the length of the latus rectum for an ellipse, and hyperbola is 2b 2/a. The length of the latus rectum for a parabola is LL' = 4a. Latus Rectum: It is a focal chord that is perpendicular to the axis of the conic.For an ellipse, hyperbola we have two foci, and hence we have two focal distances. Focal Distance: The distance of a point \((x_1, y_1)\) on the conic, from any of the foci, is the focal distance.The focal chord cuts the conic section at two distinct points. Focal Chord: The focal chord of a conic is the chord passing through the focus of the conic section.Vertex: The point on the axis where the conic cuts the axis is referred to as the vertex of the conic.Center: The point of intersection of the principal axis and the conjugate axis of the conic is called the center of the conic.The conjugate axis is also its minor axis. Conjugate Axis: The axis drawn perpendicular to the principal axis and passing through the center of the conic is the conjugate axis.Principal Axis: The axis passing through the center and foci of a conic is its principal axis and is also referred to as the major axis of the conic.The following are the details of the parameters of the conic section. Let us briefly learn about each of these parameters related to the conic section. Other than these three parameters, conic sections have a few more parameters like principal axis, latus rectum, major and minor axis, focal parameter, etc. The value of e for different conic sections is as follows. As eccentricity increases, the conic section deviates more and more from the shape of the circle. If two conic sections have the same eccentricity, they will be similar. It is a non-negative real number. Eccentricity is denoted by "e". Eccentricity is used to uniquely define the shape of a conic section. The eccentricity of a conic section is the constant ratio of the distance of the point on the conic section from the focus and directrix. The parabola has 1 directrix, the ellipse and the hyperbola have 2 directrices each. The directrix is parallel to the conjugate axis and the latus rectum of the conic. Every point on the conic is defined by the ratio of its distance from the directrix and the foci. The directrix is a line drawn perpendicular to the axis of the referred conic. Directrixĭirectrix is a line used to define the conic sections. The hyperbola has two foci and the absolute difference of the distance of the point on the hyperbola from the two foci is constant. For parabola, it is a limiting case of an ellipse and has one focus at a distance from the vertex, and another focus at infinity. Circle, which is a special case of an ellipse, has both the foci at the same place and the distance of all points from the focus is constant. For an ellipse, the sum of the distance of the point on the ellipse from the two foci is constant. A parabola has one focus, while ellipses and hyperbolas have two foci. They are specially defined for each type of conic section. The focus or foci(plural) of a conic section is/are the point(s) about which the conic section is created. Let us learn in detail about each of them. And the shape and orientation of these shapes are completely based on these three important features. The various conic figures are the circle, ellipse, parabola, and hyperbola. The focus, directrix, and eccentricity are the three important features or parameters which defined the conic.
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