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.
Thus, the standard equation of the horizontal parabola is .
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 .
- 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 , shown in the figure, is the latus rectum of the parabola .
- 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 . Thus, . Therefore, the end points of the latus rectum are .
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:
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
Condition for the General Equation of Second Degree to Represent a Parabola
The general equation of second degree
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