The eccentricity of a parabola is equal to 1 (e = 1). Hence, the derived standard equation of the parabola is y 2 = 4ax. Here the point 'M' is the foot of the perpendicular from the point P, on the directrix. Let us consider a point P(x, y) on the parabola, and using the formula PF = PM, we can find the equation of the parabola. A parabola is the locus of a point that is equidistant from a fixed point called the focus (F), and the fixed-line is called the Directrix (x + a = 0). The equation of the parabola can be derived from the basic definition of the parabola. The vertex of the parabola having the equation y 2 = 4ax is (0,0), as it cuts the axis at the origin. The vertex of the parabola is the point where the parabola cuts through the axis. The focus of the parabola is F(a, 0), and the equation of the directrix of this parabola is x + a = 0. The axis of the parabola is the x-axis which is also the transverse axis of the parabola. The standard equation of a parabola is y 2 = 4ax. Hence learning the properties and applications of a parabola is the foundation for physicists. Many of the motions in the physical world follow a parabolic path. It is the locus of a point that is equidistant from a fixed point, called the focus, and the fixed-line is called the directrix. Parabola is an important curve of the conic section. The parametric coordinates represent all the points on the parabola.įAQs on Parabola What is Parabola in Conic Section? Parametric Coordinates: The parametric coordinates of the equation of a parabola y 2 = 4ax are (at 2, 2at). For a pole having the coordinates \((x_1, y_1)\), for a parabola y 2 =4ax, the equation of the polar is \(yy_1 = 2x(x + x_1)\). ![]() And this referred point is called the pole. Pole and Polar: For a point lying outside the parabola, 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. For a point \((x_1, y_1)\) outside the parabola, the equation of the chord of contact is \(yy_1 = 2x(x + x_1)\). \(\sqrt(x - x_1)\)Ĭhord of Contact: The chord drawn to joining the point of contact of the tangents drawn from an external point to the parabola is called the chord of contact. We need to derive the equation of parabola using PF = PB The equation of the directtrix is x + a = 0 and we use the perpendicular distance formula to find PB. The coordinates of the focus is F(a,0) and we can use the coordinate distance formula to find its distance from P(x, y) Here we consider a point B on the directrix, and the perpendicular distance PB is taken for calculations.Īs per this definition of the eccentricity of the parabola, we have PF = PB (Since e = PF/PB = 1) As per the definition of a parabola, the distance of this point from the focus F is equal to the distance of this point P from the Directrix. Let us consider a point P with coordinates (x, y) on the parabola. The eccentricity of a parabola is equal to 1. It is the ratio of the distance of a point from the focus, to the distance of the point from the directrix. The endpoints of the latus rectum are (a, 2a), (a, -2a). The length of the latus rectum is taken as LL' = 4a. Latus Rectum: It is the focal chord that is perpendicular to the axis of the parabola and is passing through the focus of the parabola.The focal distance is also equal to the perpendicular distance of this point from the directrix. Focal Distance: The distance of a point \((x_1, y_1)\) on the parabola, from the focus, is the focal distance.The focal chord cuts the parabola at two distinct points. Focal Chord: The focal chord of a parabola is the chord passing through the focus of the parabola.The directrix is perpendicular to the axis of the parabola. Directrix: The line drawn parallel to the y-axis and passing through the point (-a, 0) is the directrix of the parabola.Focus: The point (a, 0) is the focus of the parabola. ![]() Some of the important terms below are helpful to understand the features and parts of a parabola. ![]() The standard equation of a regular parabola is y 2 = 4ax. The general equation of a parabola is: y = a(x-h) 2 + k or x = a(y-k) 2 +h, where (h,k) denotes the vertex. Parabola is an important curve of the conic sections of the coordinate geometry. A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola. Also, an important point to note is that the fixed point does not lie on the fixed line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed line.
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