Question

If the coordinates of a variable point 'P' be $ \left( t+\dfrac{1}{t},t-\dfrac{1}{t} \right) $ , where 't' is a
variable quantity, then find the locus of 'P'.

Answer 1

Hint:Write two equations of x and y in terms of t, and try to eliminate t from them by performing addition/subtraction or other mathematical operations.
The final equation in terms of the variables x and y is the required locus of the point P.
Observe that $ {{(a+b)}^{2}}-{{(a-b)}^{2}}=4ab $ . How can we use this to help eliminate it?

Complete step by step solution:
The coordinates of the point P are given to be $ \left( t+\dfrac{1}{t},t-\dfrac{1}{t} \right) $ .
Therefore:
$ x=t+\dfrac{1}{t} $ ... (1)
$ y=t-\dfrac{1}{t} $ ... (2)

Squaring both the equations, we get:
$ {{x}^{2}}={{\left( t+\dfrac{1}{t} \right)}^{2}} $
⇒ $ {{x}^{2}}={{t}^{2}}+2\times t\times \dfrac{1}{t}+\dfrac{1}{{{t}^{2}}} $
⇒ $ {{x}^{2}}={{t}^{2}}+\dfrac{1}{{{t}^{2}}}+2 $ ... (3)

And, $ {{y}^{2}}={{\left( t-\dfrac{1}{t} \right)}^{2}} $
⇒ $ {{y}^{2}}={{t}^{2}}-2\times t\times \dfrac{1}{t}+\dfrac{1}{{{t}^{2}}} $
⇒ $ {{y}^{2}}={{t}^{2}}+\dfrac{1}{{{t}^{2}}}-2 $ ... (4)

Subtracting equation (4) from equation (3), we get:

$ {{x}^{2}}-{{y}^{2}}=4 $ , which is the required locus of the point P.

Note:It can also be added that the equation $ {{x}^{2}}-{{y}^{2}}=4 $ represents a hyperbola with vertices at $ (\pm 2,\ 0) $ and center at the origin $ (0,\ 0) $ .
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

In other words, the set of the points that satisfy some property is often called the locus of a point satisfying this property.

e.g. The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points.

A parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object, such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.