**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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The equation Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 is the basic equation for the conic. If it is a conic, then it is

(i) **a parabola if B ^{2 }**

(ii) an ellipse if B^{2} − 4AC < 0

(iii) a hyperbola if B^{2} − 4AC > 0

**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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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) y^{2 }= 4x

**Solution:**

The basic form of parabola is y^{2}=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) x^{2 }= − 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.