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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.

Thus, the standard equation of the horizontal parabola is ${y}^{2}=4ax$.

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 ${y}^{2}=4ax$.
• 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 ${L}_{1}{L}_{2}$, shown in the figure, is the latus rectum of the parabola ${y}^{2}=4ax$.
• 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 ${y}^{2}=4ax$. 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:

 ${\left(y-k\right)}^{2}=4a\left(x-h\right)$
 ${\left(x-h\right)}^{2}=4b\left(y-k\right)$

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 ${y}^{2}=4ax$ is . Thus, any point on the parabola ${y}^{2}=4ax$ can be taken as 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 $a{x}^{2}+2hxy+b{y}^{2}+2\mathrm{g\dots }$

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