question_answer

An ellipse slides between two lines at right angle to one another. The locus of its centre is

A)  a circle                

B)  a parabola        

C)  an ellipse           

D)  a hyperbola

Correct Answer: A

Solution :

 Let the two lines be taken as \[x\] and y axes, respectivley. Let \[C(\alpha ,\,\beta )\] be the centre of the ellipse of which \[{{S}_{1}}\,({{x}_{1}},\,\,{{y}_{1}})\] and \[{{S}_{2}}\,({{x}_{2}},\,{{y}_{2}})\] are foci. Then,    \[{{x}_{1}}+{{x}_{2}}=2\alpha \] and        \[{{y}_{1}}+{{y}_{2}}=2\beta \] Since, x-axis and y-axis are the tangents to the ellipse. \[\therefore \]  \[{{x}_{1}}{{x}_{2}}={{b}^{2}}\] \[{{y}_{1}}{{y}_{2}}={{b}^{2}}\] Also,      \[{{S}_{1}}S_{2}^{2}\,=4{{a}^{2}}{{e}^{2}}\] \[\Rightarrow \]               \[{{({{x}_{1}}-{{x}_{2}})}^{2}}+{{({{y}_{1}}-{{y}_{2}})}^{2}}=4{{a}^{2}}{{e}^{2}}\] \[\Rightarrow \]               \[{{({{x}_{1}}+{{x}_{2}})}^{2}}+{{({{y}_{1}}+{{y}_{2}})}^{2}}-4{{x}_{1}}{{x}_{2}}\] \[-4{{y}_{1}}{{y}_{2}}=4{{a}^{2}}{{e}^{2}}\] \[\Rightarrow \] \[4{{\alpha }^{2}}+4{{\beta }^{2}}-4{{b}^{2}}-4{{b}^{2}}=4{{a}^{2}}{{e}^{2}}\] \[\Rightarrow \] \[{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}{{e}^{2}}=2{{b}^{2}}\] \[\Rightarrow \] \[{{\alpha }^{2}}+{{\beta }^{2}}=2{{b}^{2}}+{{a}^{2}}-{{b}^{2}}\] \[[\because \,{{b}^{2}}={{a}^{2}}\,\,(1-{{e}^{2}})]\] \[\Rightarrow \]               \[{{\alpha }^{2}}+{{\beta }^{2}}={{a}^{2}}+{{b}^{2}}\] Locus \[(\alpha ,\,\beta )\] is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}+{{b}^{2}}\] Which represents a circle.