Question

The locus of the centre of the circle x²+y²+4x costhita-2y sinthita - 10 = 0 is
(a) An ellipse
(b) A circle
(c) A hyperbola
(D) A parabola

Answer 1

$$\begin{gathered} theequationcircle{x}^2+y^2+4x\cos \theta -2y\sin \theta -10=0 \\ comparingwithgeneralequationofcircleisg=2\cos \theta andf=-\sin \theta \\ coordinateofcircle=(-g,-f) \\ =(-2\cos \theta ,\sin \theta ) \\ letthiscenter=(h,k) \\ soh=-2\cos \theta ....................\left(i\right) \\ k=\sin \theta ..............................\left(ii\right) \\ eliminating\theta from\left(i\right)and\left(ii\right) \\ {\sin }^2\theta +{\cos }^2\theta =1 \\ \Rightarrow k^2+(-h/2{)}^2=1 \\ whichislocusofcentreandrepresentscircle. \end{gathered}$$