Suppose f (t) and g(t) are functions of 't '. Then the equations x = f (t) and y = g(t) together describe a curve in the plane .

**Parametric form of Conics**

Suppose *f *(*t*) and *g*(*t*) are functions
of '*t *'. Then the equations *x *= *f *(*t*) and *y *=
*g*(*t*) together describe a curve in the plane . In general '*t *'
is simply an arbitrary variable, called in this case a **parameter**, and this method of
specifying a curve is known as **parametric equations**. One important
interpretation of '*t *' is time .
In this interpretation, the equations *x *= *f *(*t*) and *y *=
*g*(*t*) give the position of an object at time '*t *' .

So a parametric equation simply has a third variable, expressing *x
*and *y *in terms of that third variable as a parameter . A parameter
does not always have to be '*t *' . Using '*t *' is more standard but
one can use any other variable.

Let *P*(*x*, *y*) be any point on the circle *x*^{2}
+ *y*^{2} = *a*^{2}.

Join *OP *and let it make an angle *θ *with *x *-axis.

Draw *PM *perpendicular to *x *-axis. From triangle *OPM
*,

*x *= *OM *= *a *cos*θ*

*y *= *MP *= *a *sin*θ*

Thus the coordinates of any point on the given circle are (*a *cos
θ, *a *sin θ) and

*x *= *a *cos*θ *, *y *= *a *sin*θ *, 0 ≤ *θ
*≤ 2*π *are the parametric equations of the circle *x*^{2}
+ *y*^{2} = *a*^{2}.

Conversely, if *x *= *a *cos*θ *, *y *= *a *sin*θ *, 0 ≤ *θ *≤ 2*π *,

Squaring and adding, we get,

Thus *x*^{2} + *y*^{2} = *a*^{2}
yields the equation to circle with centre (0, 0) and radius *a *units.

**Note**

(1)* x *= *a *cos *t*,
*y *= *a *sin *t *, 0 ≤ *t *≤ 2*π *also represents the
same parametric equations of circle *x*^{2} + *y*^{2}
= *a*^{2} ,

*t *increasing in anticlockwise direction.

(2)* x *= *a *sin *t*,
*y *= *a *cos *t*, 0 ≤ *t *≤ 2*π *also represents the
same parametric equations of circle *x*^{2} + *y*^{2}
= *a*^{2},

*t *increasing in clockwise direction.

Let *P*(*x*_{1} , *y*_{1}) be a
point on the parabola

y_{1}^{2} = 4ax_{1}

Parametric form of *y*^{2} = 4*ax *is *x *=
*at*^{2}, *y *= 2*at*, −∞ < *t *< ∞ .

Conversely if *x *= *at*^{2} and *y *= 2*at*,
−∞< *t *< ∞ , then eliminating '*t *' between these equations
we get *y*^{2} = 4*ax *.

Let *P *be any point on the ellipse. Let the ordinate *MP *meet
the auxiliary circle at *Q *.

Let ∠ *ACQ *= *α*

∴ *CM *=
*a *cos*α *, *MQ *= *a *sin *α*

and *Q*(*a *cos*α *, *a *sin *α *)

Now *x *-coordinate of *P *is *a *cos*α *.
If its *y *-coordinate is *y*′, then *P*(*a *cos*α *, *y*′)
lies on

Hence *P *is (*a *cos*α *, *b *sin *α *)
.

The parameter *α *is called the eccentric angle of the point *P
*. Note that *α *is the angle which the line *CQ *makes with the *x
*-axis and not the angle which the line *CP *makes with it.

Hence the parametric equation of an ellipse is *x *= *a *cos*θ
*, *y *= *b *sin*θ *, where *θ *is the parameter 0 ≤ *θ *≤ 2*π *.

Similarly, parametric equation of a hyperbola can be derived as *x
*= *a *sec*θ *, *y *= *b *tan*θ *, where *θ
*is the parameter. −*π *≤ *θ *≤ *π *except θ = ± π/2.

In nutshell the parametric equations of the circle,
parabola,ellipse and hyperbola are given in the following table.

(1) Parametric form represents a family of
points on the conic which is the role of a parameter. Further parameter plays
the role of a constant and a variable, while cartesian form represents the
locus of a point describing the conic. Parameterisation denotes the orientation
of the curve.

(2) A parametric representation need not be
unique.

(3) Note that using parameterisation reduces
the number of variables at least by one.

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12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II : Parametric form of Conics |

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