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 x2 + y2 = a2.
Join OP and let it make an angle θ with x -axis.
Draw PM perpendicular to x -axis. From triangle OPM ,
x = OM = a cosθ
y = MP = a sinθ
Thus the coordinates of any point on the given circle are (a cos θ, a sin θ) and
x = a cosθ , y = a sinθ , 0 ≤ θ ≤ 2π are the parametric equations of the circle x2 + y2 = a2.
Conversely, if x = a cosθ , y = a sinθ , 0 ≤ θ ≤ 2π ,
Squaring and adding, we get,
Thus x2 + y2 = a2 yields the equation to circle with centre (0, 0) and radius a units.
(1) x = a cos t, y = a sin t , 0 ≤ t ≤ 2π also represents the same parametric equations of circle x2 + y2 = a2 ,
t increasing in anticlockwise direction.
(2) x = a sin t, y = a cos t, 0 ≤ t ≤ 2π also represents the same parametric equations of circle x2 + y2 = a2,
t increasing in clockwise direction.
Let P(x1 , y1) be a point on the parabola
y12 = 4ax1
Parametric form of y2 = 4ax is x = at2, y = 2at, −∞ < t < ∞ .
Conversely if x = at2 and y = 2at, −∞< t < ∞ , then eliminating 't ' between these equations we get y2 = 4ax .
Let P be any point on the ellipse. Let the ordinate MP meet the auxiliary circle at Q .
Let ∠ ACQ = α
∴ CM = a cosα , MQ = a sin α
and Q(a cosα , a sin α )
Now x -coordinate of P is a cosα . If its y -coordinate is y′, then P(a cosα , y′) lies on
Hence P is (a cosα , 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 cosθ , y = b sinθ , where θ is the parameter 0 ≤ θ ≤ 2π .
Similarly, parametric equation of a hyperbola can be derived as x = a secθ , y = b tanθ , 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.