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If a \: \: and\: \: c  are positive real numbers and the ellipse  \frac{x^{2}}{4c^{2}}+\frac{y^{2}}{c^{2}}=1  has four distinct points in common with the circle x^{2}+y^{2}=9a^{2}  , then 

Option: 1

9ac-9a^{2}-2c^{2}<0


Option: 2

6ac+9a^{2}-2c^{2}<0


Option: 3

9ac-9a^{2}-2c^{2}>0


Option: 4

6ac+9a^{2}-2c^{2}>0


Answers (1)

best_answer

Given equation of the ellipse is \frac{x^{2}}{4c^{2}}+\frac{y^{2}}{c^{2}}=1 and equation of the circle is x^2+y^2=9a^2

radius = 3a

major axis = 4c

Now, radius < Half of the length of the major axis.

\\3 a<2 c \\ 9 a^{2}<4 c^{2} \\ 9 a c-9 a^{2}>9 a c-4 c^{2}\\ 9 a c-9 a^{2}-2 c^{2}>9 a c-6 c^{2} \\ \text { Again } 3 a<2 c \\ \Rightarrow 9 a c<6 c^{2} \\ \Rightarrow 9 a c-6 c^{2}<0\\ \text { From above}\\ 9 a c-9 a^{2}-2 c^{2}>0

Posted by

Rakesh

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