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  •  A variable circle has the equation <img alt="\mathrm{x}^{2}+\mathrm{y}^{2}-2\left(\mathrm{t}^{2

 A variable circle \mathrm{C} has the equation \mathrm{x}^{2}+\mathrm{y}^{2}-2\left(\mathrm{t}^{2}-3 \mathrm{t}+1\right) \mathrm{x}-2\left(\mathrm{t}^{2}+2 \mathrm{t}\right) \mathrm{y}+\mathrm{t}=0 \text {, where } \mathrm{t} \text { is a parameter. }
The locus of the centre of the circle is

Option: 1

a parabola


Option: 2

an ellipse


Option: 3

a hyperbola


Option: 4

pair of straight lines


Answers (1)

best_answer

\mathrm{centre \,is\, \mathrm{x}=\mathrm{t}^{2}-3 \mathrm{t}+1}\quad \ldots(1)
                   \mathrm{y=t^{2}+2 t}\quad \ldots(2) 

eliminating t, we get

\mathrm{\mathrm{x}=\mathrm{t}^{2}+2 \mathrm{t}-5 \mathrm{t}+1=\mathrm{y}-5 \mathrm{t}+1}
\mathrm{\mathrm{t}=\frac{\mathrm{y}-\mathrm{x}+1}{5}}
Substituting the value of \mathrm{t} in (2)
\mathrm{\mathrm{y}=\left(\frac{\mathrm{y}-\mathrm{x}+1}{5}\right)^{2}+2\left(\frac{\mathrm{y}-\mathrm{x}+1}{5}\right)}
\mathrm{25 \mathrm{y}=(\mathrm{y}-\mathrm{x}+1)^{2}+10(\mathrm{y}-\mathrm{x}+1)}
\mathrm{25 y=y^{2}+x^{2}+1-2 x y-2 x+2 y+10 y-10 x+10}
\mathrm{x^{2}+y^{2}-2 x y-12 x-13 y+11=0}

which is a parabola
\mathrm{as \: \Delta \neq 0 \: and\: \mathrm{h}^{2}=\mathrm{ab}}
 

Posted by

sudhir kumar

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