An ellipse is drawn by taking a diameter of thec circle \[{\left( {x - 1} \right)^2} + {y^2} = 1\] as its semi-minor axis and a diameter of the circle \[{x^2} + {\left( {y - 2} \right)^2} = 4\] is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :
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Official solution
Equation of circle is \[{\left( {x - 1} \right)^2} + {y^2} = 1\]
\[ \Rightarrow \] radius =1 and diameter =2
\[\therefore\] Length of semi-minor axis is 2.
Equation of circle is \[{x^2} + {\left( {y - 2} \right)^2} = 4 = {\left( 2 \right)^2}\]
\[ \Rightarrow \] radius =2 and diameter =4
\[\therefore\] Length of semi major axis is 4
We know, equation of ellipse is given by
\[{{{x^2}} \over {\left( {Major\,\,\,axi{s^{\,\,2}}} \right)}} + {{{y^2}} \over {\left( {Minor\,\,\,axi{s^{\,\,2}}} \right)}} = 1\]
\[ \Rightarrow {{{x^2}} \over {{{\left( 4 \right)}^2}}} + {{{y^2}} \over {{{\left( 2 \right)}^2}}} = 1\]
\[ \Rightarrow {{{x^2}} \over {16}} + {{{y^2}} \over 4} = 1\]
\[ \Rightarrow {x^2} + 4{y^2} = 16\]