JEE MAIN - Mathematics (2012 - No. 15)
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 :
$$4{x^2} + {y^2} = 4$$
$${x^2} + 4{y^2} = 8$$
$$4{x^2} + {y^2} = 8$$
$${x^2} + 4{y^2} = 16$$
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
$${{{x^2}} \over {\left( {Major\,\,\,axi{s^{\,\,2}}} \right)}} + {{{y^2}} \over {\left( {Minor\,\,\,axi{s^{\,\,2}}} \right)}} = 1$$
$$ \Rightarrow {{{x^2}} \over {{{\left( 4 \right)}^2}}} + {{{y^2}} \over {{{\left( 2 \right)}^2}}} = 1$$
$$ \Rightarrow {{{x^2}} \over {16}} + {{{y^2}} \over 4} = 1$$
$$ \Rightarrow {x^2} + 4{y^2} = 16$$
$$ \Rightarrow $$ 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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