x={a(1-t²)}{1+t²}, y={2at}{1+t²} What does the equation represent?

Correct Answer is : It represents a circle of radius a

It represents a circle of diameter a

Correct Answer is : It represents a circle of radius a

Test Index

UPSC NDA I 2015 Mathematics Paper

Question Numbers: 82-84

For the next three (3) items that follow:

Consider the parametric equation

Question : 82 of 120

Marks: +1, -0

x=\frac{a(1-t^{2})}{1+t^{2}}, y=\frac{2at}{1+t^{2}}

What does the equation represent?

Solution:

Concept:

The parametric equations define a curve; eliminating the parameter

t

gives the Cartesian equation, which in this case is a circle.

Explanation:

Given

x = \frac{a(1-t^{2})}{1+t^{2}}

and

y = \frac{2at}{1+t^{2}}

Compute

x^{2} + y^{2}

x^{2} = \frac{a^{2}(1 - 2t^{2} + t^{4})}{(1+t^{2})^{2}}

y^{2} = \frac{4a^{2}t^{2}}{(1+t^{2})^{2}}

Add them:

x^{2}+y^{2} = \frac{a^{2}(1 - 2t^{2} + t^{4} + 4t^{2})}{(1+t^{2})^{2}} = \frac{a^{2}(1 + 2t^{2} + t^{4})}{(1+t^{2})^{2}}

Factor numerator:

1 + 2t^{2} + t^{4} = (1+t^{2})^{2}

Thus

x^{2}+y^{2} = \frac{a^{2}(1+t^{2})^{2}}{(1+t^{2})^{2}} = a^{2}

This is the equation of a circle with centre at

(0,0)

and radius

a

Answer:

It represents a circle of radius

a

. So option B is correct.

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