KVPY SX SB Exam 20141 Question Paper

Question : 86 of 120

Marks: +1, -0

An ellipse inscribed in a semi-circle touches the circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. When the ellipse has the maximum possible area.
its eccentricity is

$1/√2$
$1/2$
$1/ √3$
$√{2/3}$

Validate

Solution:

Let ellipse

x^2/a^2+y^2/b^2=1

and circle

x^2 + (y + b)^2 = r^ 2

{let radius = r}
put

x^2 = a^2-{a^2y^2}/b^2

in circle

a^2-{a^2y^2}/b^2+ (y + b)^2 = r^ 2

⇒

(1-a^2/b^2)y ^2 + 2by + (a^2 + b^2 -r ^2 ) = 0

D = 0 ⇒ r^ 2 =a^4/{a^2-b^2}

⇒

b = a√{1-a^2/r^2}

Area

= ∆ = πab = πa^2√{1-a^2/r^2}

{d∆ }/{da}=0

⇒a^2={2r^2}/3⇒a=√{2/3}r

∴ b =a√{1-2/3}=a/√3

⇒e=√{2/3}

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