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1)

 The ellipse   E1:x29+y24=1 is  inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point (0, 4)circumscribes the rectangle R. The eccentricity of the ellipse E2 is 


A) 22

B) 32

C) 12

D) 34

Answer:

Option C

Explanation:

Concept involved equation of an ellipse is

x2a2+y2b2=1    (a >b)

 Ecentricity  e2=1b2a2       (a >b)

 2411202139_l2.PNG

 

 

Description of Situation As ellipse circumscribes the rectangle, then it must
pass through all four vertices.

Sol. Let the equation of Ellipse  E2 be 

 x2a2+y2b2=1 , where a<b and b=4

 24112021113_l3.PNG

 

 Also, it passes through (3,2) 

    9a2+4b2=1                     ( b=4)

 \Rightarrow  \frac{9}{a^{2}}+\frac{1}{4}=1 or  a^{2}=12

 Eccentricity of E_{2}

 \Rightarrow   e^{2}=1-\frac{a^{2}}{b^{2}}=1-\frac{12}{16}=\frac{1}{4}     ( \because a <b)

 \therefore  e=\frac{1}{2}