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Conics

Equation of a Parabola

Standard Equation of the Parabola A parabola is defined as the locus of a point P equidistant from a fixed point (called focus) and a fixed line (called directrix).  Let us consider origin (0, 0) as the vertex A of the parabola and let P (x, y) be any point on the locus. Let the fixed point (focus) be S (a, 0) and the fixed line be x + a = 0. Using the definition of parabola, we have PS = PM. PS2=PM2 ⇒x-a2+y2=x+a2 ⇒y2=4ax Thus, the standard equation of the horizontal parabola is y2=4ax. Important Points and Lines Related to Parabola The constant ratio is called the eccentricity and is denoted by e. When the eccentricity is unity; e = 1, the conic is called a Parabola. ​ The line which passes through the focus and perpendicular to the directrix is called axis of the parabola. Here x-axis is the axis of the parabola y2=4ax. The vertex of a parabola is defined as the intersection point of the parabola and its axis. Here (0, 0) is the vertex. The chord passing through the focus and perpendicular to the axis is called latus rectum. The chord L1L2, shown in the figure, is the latus rectum of the parabola y2=4ax. Any chord which is perpendicular to the axis of the parabola is called double ordinate. The straight line passing through the vertex and perpendicular to the axis of the parabola is called tangent at vertex. Here, y-axis is the tangent at the vertex (0, 0). To find the end points of the latus rectum, put x = a in y2=4ax. Thus, y2=4a2 ⇒y=±2a. Therefore, the end points of the latus rectum are L1a, 2a and L2a,-2a. Other Horizontal and Vertical Parabolas The following parabolas are drawn with their vertex, focus, etc.                                           Equation of the Parabola in Non-standard Form   The horizontal and vertical parabolas with vertex other than the origin are:   y-k2=4ax-h     x-h2=4by-k                                                          Equation of the Parabola in Non-standard Form                 The details regarding focus, vertex, directrix, etc. are:        (y − k)2 = 4a (x − h) (x − h)2 = 4b (y − k) Vertex (h, k) (h, k) Focus (a + h, k) ( h, b + k) Equation of the Directrix (x − h) + a = 0 (y − k) + b = 0 Equation of the axis y = k x = h Tangent at the vertex x = h y = k Equation of latus rectum x − a = h y − k = b Length of latus rectum |4a| |4b| End points of latus rectum L1 (a + h, 2a + k) and L2 (a + h,  −2a + k) L1( −2b + h, b + k) and L2 (2b + h,  b + k) Parametric Equation of a Parabola   The parametric equation of the standard parabola y2=4ax is x=at2, y=2at. Thus, any point on the parabola y2=4ax can be taken as at2, 2at and we call this point as point t. Condition for the General Equation of Second Degree to Represent a Parabola The general equation of second degree ax2+2hxy+by2+2gx+2fy+c=0 represents a parabola, if abc+2fgh-af2-bg2-ch2≠0 and h2=ab. Focal Distance of a Point Focal distance of a point P on parabola is defined as the distance between the point P and its focus S. Here, PS is the focal distance. ∴ PS = PM = AN + AS = a + x = a + at2 = a (1 + t2) Position of a Point Relative to a Parabola Let the standard parabola be y2=4ax. Take two points P1x1, y1 and P2x2, y2 outside and inside the parabola as shown in the figure. Then: For point P1x1, y1, we have y12-4ax1>0 For point P2x2, y2, we have y22-4ax2<0 Standard Equation of the Parabola A parabola is defined as the locus of a point P equidistant from a fixed point (called focus) and a fixed line (called directrix).  Let us consider origin (0, 0) as the vertex A of the parabola and let P (x, y) be any point on the locus. Let the fixed point (focus) be S (a, 0) and the fixed line be x + a = 0. Using the definition of parabola, we have PS = PM. PS2=PM2 ⇒x-a2+y2=x+a2 ⇒y2=4ax Thus, the standard equation of the horizontal parabola is y2=4ax. Important Points and Lines Related to Parabola The constant ratio is called the eccentricity and is denoted by e. When the eccentricity is unity; e = 1, the conic is called a Parabola. ​ The line which passes through the focus and perpendicular to the directrix is called axis of the parabola. Here x-axis is the axis of the parabola y2=4ax. The vertex of a parabola is defined as the intersection point of the parabola and its axis. Here (0, 0) is the vertex. The chord passing through the focus and perpendicular to the axis is called latus rectum. The chord L1L2, shown in the figure, is the latus rectum of the parabola y2=4ax. Any chord which is perpendicular to the axis of the parabola is called double ordinate. The straight line passing through the vertex and perpendicular to the axis of the parabola is called tangent at vertex. Here, y-axis is the tangent at the vertex (0, 0). To find the end points of the latus rectum, put x = a in y2=4ax. Thus, y2=4a2 ⇒y=±2a. Therefore, the end points of the latus rectum are L1a, 2a and L2a,-2a. Other Horizontal and Vertical Parabolas The following parabolas are drawn with their vertex, focus, etc.                                           Equation of the Parabola in Non-standard Form   The horizontal and vertical parabolas with vertex other than the origin are:   y-k2=4ax-h     x-h2=4by-k                                                          Equation of the Parabola in Non-standard Form                 The details regarding focus, vertex, directrix, etc. are:        (y − k)2 = 4a (x − h) (x − h)2 = 4b (y − k) Vertex (h, k) (h, k) Focus (a + h, k) ( h, b + k) Equation of the Directrix (x − h) + a = 0 (y − k) + b = 0 Equation of the axis y = k x = h Tangent at the vertex x = h y = k Equation of latus rectum x − a = h y − k = b …

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