GATE Electronics and Communications (EC) 2020 Solved Papers

Question : 8 of 65

Marks: +1, -0

A circle with centre

{O}

is shown in the figure. A rectangle PQRS of maximum possible area is inscribed in the circle. If the radius of the circle is a, then the area of the shaded portion is...................

$π a^{2}-3 a^{2}$
$π a^{2}-2 a^{2}$
$π a^{2}-a^{2}$
$π a^{2}-√{2} a^{2}$

Validate

Solution:

r^{2}=\;{a^{2}}/{4}+\;{b^{2}}/{4}

∵ \;{b^{2}}/{4}=r^{2}-\;{a^{2}}/{4}

b=± √{4 r^{2}-a^{2}}

b^{2}=4 r^{2}-a^{2}

\;\text " Area "\; A=a b

=± a √{4 r^{2}-a^{2}}

A^{2}=a^{2}(4 r^{2}-a^{2})

\;{ {d} {A}^{2}}/{ {da}} 4 r^{2} × 2 {a}-4 {a}^{3}=0

4 {a}(2 {r}^{2}- {a}^{2})=0

⇒ {a}^{2}=2 {r}^{2}

∴ {b}^{2}=4 {r}^{2}-2 {r}^{2}

=2 r^{2}

∴

area of rectangle

=(r √{2})^{2}

=2 r^{2}

area of circle

=n r^{2}

∴

Required area

=(Π-2) r^{2}

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