Q.134. Parametric equations of a curve. Point at which the curve intersects itself.

Question 134.
A curve is defined by parametric equations:
x(t) = t^3 - 3t
y(t) = t^2 - t
If a graph of y vs. x is plotted, then the curve intersects itself at a point. What are the coordinates of that point?

Answer 134.
The curve is defined by parametric equations:
x(t) = t^3 - 3t
y(t) = t^2 - t

Let the curve intersect itself for t = t1 and t = t2
=> x(t1) = x(t2) and y(t1) = y(t2)

Now, x(t1) = x(t2)
=> t1^3 - 3t1 = t2^3 - 3t2
=> t1^3 - t2^3 = 3(t1 - t2)
=> t1^2 + t1t2 + t2^2 = 3 ... (1)

and, y(t1) = y(t2)
=> t1^2 - t1 = t2^2 - t2
=> t1^2 - t2^2 = t1 - t2
=> t1 + t2 = 1 ... (2)

Plugging t2 = 1- t1 from (2) in (1),
t1^2 + t1(1 - t1) + (1 - t1)^2 = 3
=> t1^2 - t1 - 2 = 0
=> (t1 - 2)(t1 + 1) = 0
=> t1 = 2 or t1 = -1 and corresponding
t2 = -1 or t2 = 2

=> Point of intersection, by putting t1 = 2, is
x = (2)^3 - 3(2) = 2 and
y = (2)^2 - 2 = 2

t2 = -1 gives the same point as can be verified
x = (-1)^3 - 3(-1) = 2 and
y = (-1)^2 - (-1) = 2.

Answer : (2, 2).

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