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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 :

JEE · Math · previous-year question

  1. A.$$4{x^2} + {y^2} = 4$$
  2. B.$${x^2} + 4{y^2} = 8$$
  3. C.$$4{x^2} + {y^2} = 8$$
  4. D.$${x^2} + 4{y^2} = 16$$correct

Answer

D. $${x^2} + 4{y^2} = 16$$

Explanation

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 $${{{x^2}} \over {\left( {Major\,\,\,axi{s^{\,\,2}}} \right)}} + {{{y^2}} \over {\left( {Minor\,\,\,axi{s^{\,\,2}}} \right)}} = 1$$ $$ \Rightarrow {{{x^2}} \over {{{\left( 4 \right)}^2}}} + {{{y^2}} \over {{{\left( 2 \right)}^2}}} = 1$$ $$ \Rightarrow {{{x^2}} \over {16}} + {{{y^2}} \over 4} = 1$$ $$ \Rightarrow {x^2} + 4{y^2} = 16$$

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