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Parabola Std form

Standard form of parabola

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (i.e. focus) is always equal to its distance from a fixed straight line (directrix). A parabola is a graph of a quadratic function, such as f(x)=x^2 .

Standard Parabola

The general form of standard parabola is: y^2=4ax , where a is a constant.

Important terms

  • The straight line passing through the focus and perpendicular to the directrix is called the Axis of the Parabola. The parabola is a symmetrical about its axis (i.e. y=0 line). It means if (x,y) lies on parabola. then (x,-y) also lies on other sides of the axis.
  • The point which bisects every chord of the conic passing through it is called the Centre of the parabola.
  • The points of intersection of the conic section and the axis called Vertex. Vertex is point O(0,0) i.e. the origin is the vertex of the parabola.
  • The fixed point is known as Focus (denoted by S).
  • The fixed line is known as Directrix.
  • A chord passing through the focus is known as focal chord.
  • The straight line through focus and perpendicular to directrix is known as Axis.
  • The focal chord which is perpendicular to the axis is known as Latus Rectum. Since it passes through the focus of parabola, the equation of latus rectum is x=a\ldots(1) .

Solving equation (1) and y^2=4ax we get: y=\pm2a .

Parametric Equations

From the equation of parabola, we can write \frac{y}{2a}=\frac{2x}{y}=t where t is a parameter. Then, y=2at and x=at^2

The equation x=at^2 and y=2at are called parametric equations.

General Form of Parabola

Finding the equation of parabola when focus and line of directrix are give Assume that the focus is S(h,k), line of directrix as ax+by+C=0 and point P as (x_1,y_1)\equiv(\alpha,\beta) whose locus is parabola. As we know that for parabola, PS=PN (since e of parabola is 1) </math>PS^2=PN^2</math>

\Rightarrow(x_1-h)^2+(y_1-k)^2=\left(\frac{(ax_1+by_1+C)^2}{\sqrt{a^2+b^2}}\right)^2

After simpligying the above equation and then replacing x_1 by x and y_1 by y , we get the required equation to parabola. The simplified form of general equation of parabola would look like:

(bx-ax)^2+2gx+2fy+c=0, Where g, f and c are real constants.

See also

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