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Show that the point (x ,y) given by x=(2...

Show that the point `(x ,y)` given by `x=(2a t)/(1+t^2) and y=((1-t^2)/(1+t^2))` lies on a circle for all real values of `t` such that `-1lt=tlt=1,` where a is any given real number.

Text Solution

Verified by Experts

Given points are `x=(2at)/(1+t^(2))` and `y=(a(1-t^(2)))/(1+t^(2))`
`because x^(2)+y^(2)=(4a^(2)t^(2))/((1+t^(2))^(2))+(a^(2)(1-t^(2)))/(1+t^(2))`
`rArr1/(a^(2)(x^(2)+y^(2)))=(4t^(2)+1+t^(4)-2t^(2))/((1+t^(2))^(2))`
`rArr1/(a^(2))(x^(2)+y^(2))=(t^(2)+2t^(2)+1)/((1+t^(2))^(2))`
`rArr 1/(a^(2))(x^(2)+y^(2))=((1+t^(2))^(2))/((1+t^(2))^(2))`
`rArr x^(2)+y^(2)=a^(2)`, which is a required circle.
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