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Question: If the coordinates of a variable point 'P' be \( \left( t+\dfrac{1}{t},t-\dfrac{1}{t} \right) \) , w...

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

Explanation

Solution

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{{(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 (t+1t,t−1t)\left( t+\dfrac{1}{t},t-\dfrac{1}{t} \right) .
Therefore:
x=t+1tx=t+\dfrac{1}{t} ... (1)
y=t−1ty=t-\dfrac{1}{t} ... (2)

Squaring both the equations, we get:
x2=(t+1t)2{{x}^{2}}={{\left( t+\dfrac{1}{t} \right)}^{2}}
⇒ x2=t2+2×t×1t+1t2{{x}^{2}}={{t}^{2}}+2\times t\times \dfrac{1}{t}+\dfrac{1}{{{t}^{2}}}
⇒ x2=t2+1t2+2{{x}^{2}}={{t}^{2}}+\dfrac{1}{{{t}^{2}}}+2 ... (3)

And, y2=(t−1t)2{{y}^{2}}={{\left( t-\dfrac{1}{t} \right)}^{2}}
⇒ y2=t2−2×t×1t+1t2{{y}^{2}}={{t}^{2}}-2\times t\times \dfrac{1}{t}+\dfrac{1}{{{t}^{2}}}
⇒ y2=t2+1t2−2{{y}^{2}}={{t}^{2}}+\dfrac{1}{{{t}^{2}}}-2 ... (4)

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

x2−y2=4{{x}^{2}}-{{y}^{2}}=4 , which is the required locus of the point P.

Note: It can also be added that the equation x2−y2=4{{x}^{2}}-{{y}^{2}}=4 represents a hyperbola with vertices at (±2, 0)(\pm 2,\ 0) and center at the origin (0, 0)(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.