Doubt from ellipse

Question 2,9th first part and 10th question both parts

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find min value just apply
\textit{cauchy schwarz inequality}

From your working for second seems answer key has given wrong for second even i had got e=(2/3)^(1/2) but answer key had given e=2/root3 thanks for confirming answer

Please explain how. I tried using it, but we require min 2 pairs of numbers right?

Take multiply and divide by \sqrt2

Question 1 and 2 both parts

Used \textbf{am-gm}
Mostly all parts r similar type think like that

  • to find min value of deno u can use calculas buts it to trapping in above also @darshil_2020 solved @abhijit_2020 us am gm will kill problm in less time
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Question 7 i haven't studied hyperbola yet is hyperbola knowledge required to do this question?

No you just use y=mx+a/m for a tangent on parabola and then substitute the value of y in xy=2 . U will get a quadratic use discriminant =0

you do need the knowledge and you do not need the knowledge about hyperbola.It depend on the method you use to solve the question

If you have the knowledge you can save time as we can directly equal the condition of tangency of two curve.
But if you do not have know about any of the one conic then this method is most welcoming ,in my opinion

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Ok thank you and help in question 6 i couldn't understand