Class 11

Math

Algebra

Sequences and Series

Assertion(A): The minimum radius vector of the curve $x_{2}a_{2} +y_{2}b_{2} =1$ is of length $a+b$

Reason (R) : The minimum value of $a_{2}sec_{2}θ+b_{2}cosec_{2}θ$ is $(a+b)_{2}$

- Both A and R are true and R is the correct explanation of A
- Both A and R are true and R is not the correct explanation of A
- A is true and R is false
- A is false and R is true

Radius vector $=r=a_{2}sec_{2}θ+b_{2}csc_{2}θ $

$⇒r_{2}=a_{2}sec_{2}θ+b_{2}csc_{2}θ=a_{2}(1+tan_{2}θ )+b_{2}(1+cot_{2}θ )=a_{2}+b_{2}+a_{2}tan_{2}θ+b_{2}cot_{2}θ$

Since, A.M$≥$G.M

Therefore, $2a_{2}tan_{2}θ+b_{2}cot_{2}θ ≥a_{2}tan_{2}θb_{2}cot_{2}θ =ab$

$⇒r_{2}≥a_{2}+b_{2}+2ab=(a+b)_{2}$

Therefore, $r≥a+b$