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Equation of Curves

In free geometry help ,General equation of curve is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

If B2 – 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

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OA = r,

{(x – a)2 + (y – b)2}1/2 = r

(x – a)2 + (y – b)2 = r2

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

(x – a)2 + (y – b)2 = r2

x2 + y2 – 2ax – 2by + (a2 + b2 – r2) = 0

Let c = a2 + b2 – r2

So the general equation of circle is

x2 + y2 – 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.

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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

x2 + y2 – 2py + p2 = y2 + 2py + p2

x2 = 4py

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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

y2 =4px

Equation of Ellipse:

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

x2/a2 + y2/b2 = 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)

x2/a2 + y2/b2 = 1

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

c = (a2 – b2)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 x2/a2 - y2/b2 = 1 or y2/b2 - x2/a2 = 1 where c = (a2 – b2)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 x2/a2 - y2/b2 = 1

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The transverse axis is in the vertical direction if the y2 term is positive and in the horizontal direction if the x2 term is positive.

  • The branches of the hyperbola open up/down if the y2 term is positive and left/right if the x2 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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