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Parabola Solver

Introduction:

In analytical geometry, a parabola is a conic section obtained on slicing a right circular cone by a planeparallel to the line joining vertex and any other point of the cone.

           

Along with focus F and vertex V, P is any point on parabola. PX is parallel to VFA. Hence the angle formed by the PX and VFA at point P is always equal.

 

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Classification of parabola with respect to the general equation of a conic:


The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is the basic equation for the conic. If it is a conic, then it is

(i)             a parabola if B2 4AC = 0

(ii)            an ellipse if B2 − 4AC < 0

(iii)           a hyperbola if B2 − 4AC > 0  


Standard equation of parabola solver:


           

Given:

  •  Fixed point (F)
  •  Fixed line (l)
  •  Eccentricity (e = 1)
  •  Moving point P(x, y)

 

Important definitions on parabolas solver:

Focus: The fixed point to draw the parabola is called the focus (F). At this point, the focus is F (a, 0).

Directrix: Thefixed line used to draw a parabola is known as the directrix of the parabola. Here, the equation of the directrix is x = − a.

Axis: Theaxis of the parabola is called the axis of symmetry. The curve y2 = 4axis symmetrical about x-axis and thus x-axis or y = 0 is the axis of theparabola y2 = 4ax. Here the axis of the parabola passes through the focus and it is perpendicular to the directrix.

Vertex: The point of intersection of a parabola and its axis is known as vertex. Here, the vertex is V (0, 0).

Focal distance: The focal distance is the distance between a point on the parabola and its focus.

Focal chord: A chord which passes through the focus of the parabola is called the focal chord of the parabola.

Latus Rectum: It is a focal chord perpendicular to the axis of the parabola. Here, the equation of the latus rectum is x="a.


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Example Problems on parabolas solver:


Let us see some of the examples on parabolas solver.

Example 1:

Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the following parabola and hence draw their graph.

(i)         y2 = 4x

Solution:

The basic form of parabola is y2=4ax.

Given equation is y2="4x." Hence a="1.

The equation of the parabola is (y − k) 2 = − 4a(x − h)

(y − 0)2 = 4(1) (x − 0)

Here (h, k) is (0, 0) and a = 1

Axis: The axis of symmetry is x-axis.

Vertex: The vertex V (h, k) is (0, 0)

Focus: The focus F (a, 0) is (1, 0)

Directrix: The equation of the directrix is x = − a i.e. x = − 1

Latus Rectum: The equation of the latus rectum is x = a i.e. x = 1 and its length is 4a = 4(1) = 4.

The graph for the above parabola is:

            

Example 2:

            Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the following parabola and hence draw their graph.

(i)          x2 = − 4y

Solution:

            (x − 0)2 = − 4(1) (y − 0)

            Here (h, k) is (0, 0) and a = 1

Axis: y-axis or x = 0

Vertex: V (0, 0)

Focus: F (0, − a) i.e. F (0, − 1)

Directrix: y = a i.e. y = 1

Latus rectum: y = − a i.e. y = − 1

                     : Length = 4

The graph for the above parabola is:

           

These are the examples of parabolas solver.