In free geometry help ,General equation of curve is Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0

If B^{2} – 4AC

< 0; then curve is ellipse

= 0; then curve is circle

> 0; then curve is hyperbola

## Equation of Circle:

In xy coordinate system let the point O (a, b) is the centre and r is the radius of the circle and there is a point A having coordinate (x, y). so

OA = r,

{(x – a)^{2} + (y – b)^{2}}^{1/2} = r

(x – a)^{2} + (y – b)^{2} = r^{2}

So the equation of a circle having centre (a, b) and radius r is

(x – a)^{2} + (y – b)^{2} = r^{2}

x^{2} + y^{2} – 2ax – 2by + (a^{2} + b^{2} – r^{2}) = 0

Let c = a^{2} + b^{2} – r^{2}

So the general equation of circle is

**x**^{2}** + y**^{2}** – 2ax – 2by + c = 0**

## Equation of Parabola:

The parabola is defined as the locus of a point which is always at the same distance from the focus and directrix.

In the following figure point F (0, p) is the focus and the directrix is the line y = -p.

Focal distance in the above figure is |p|. focal distance is the half of the distance between the directrix and focus.

Equation of parabola

{(x – 0)^{2} + (y – p)^{2}}^{1/2} = y + p

(x – 0)^{2} + (y – p)^{2 }= (y + p)^{2}

x^{2} + y^{2} – 2py + p^{2} = y^{2} + 2py + p^{2}

x^{2} = 4py

In the above figure focus is (p,0) and directrix is x = -p so the equation of parabola is

{(x – p)^{2} + (y – 0)^{2}}^{1/2 }= x + p

On simplifying

y^{2} =4px

## Equation of Ellipse:

The equation for an ellipse with a horizontal major axis is given by:

x^{2}/a^{2} + y^{2}/b^{2} = 1

The ellipse is defined as the locus of a point (*x*,*y*) which moves so that the sum of its distances from two fixed points (called *foci**,* or *focuses *) is constant.

The foci (plural of 'focus') of the ellipse (with horizontal major axis)

x^{2}/a^{2} + y^{2}/b^{2} = 1

are at (-*c*,0) and (*c*,0), where *c* is given by:

c = (a^{2} – b^{2})^{1/2}

The vertices of an ellipse are at (-*a*, 0) and (*a*, 0).

In its general form, with the origin at the center of coordinates

- the foci are at

(+/- ae, 0)

- the directrix are at

x = +/- (a/e)

- the major axis of of length 2a
- the minor axis is of length 2b

## Equation of Hyperbola:

The hyperbola is defined as the locus of a point (*x*,*y*) which moves so that the difference of its distances from two fixed points (called *foci**,* or *focuses *) is constant. The midpoint of the segment connecting the foci is called the centre of the hyperbola.

The equation of a hyperbola takes the form x^{2}/a^{2} - y^{2}/b^{2} = 1 or y^{2}/b^{2} - x^{2}/a^{2} = 1 where c = (a^{2} – b^{2})^{1/2} in both cases and locates each focus a distance of c from the center (origin) along the transverse axis. The important features for x^{2}/a^{2} - y^{2}/b^{2} = 1

The transverse axis is in the vertical direction if the y^{2} term is positive and in the horizontal direction if the x^{2} term is positive.

- The branches of the hyperbola open up/down if the y
^{2}term is positive and left/right if the x^{2}term is positive. - The slopes of the asymptotes are given by
- the foci are at (+/- ae, 0)
- the directrix are at x = +/- a/e
- the transverse axis of of length 2a

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