If p and q are positive real numbers such that p2+q2=1, If the maximum value of (p + q) is equal to √a then a is equal to

Question

If p and q are positive real numbers such that p2+q2=1, If the maximum value of (p + q) is equal to √a then a is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Given that p2+q2=1 ∴ p=cos θ and q = sin θ Then p+q=cos θ+sin θ We know that -√a2+b2 le a cos θ+b sin θ le √a2+b2 ∴ -√2 le cos θ le √2 Hence max. value of p + q is √2

Q. If $p$ and $q$ are positive real numbers such that $p^{2} + q^{2} = 1$ , If the maximum value of $(p + q)$ is equal to $a$ then a is equal to

Given that $p^{2} + q^{2} = 1$
$∴ p = cos θ$ and $q = s in θ$
Then $p + q = cos θ + s in θ$
We know that
$- a^{2} + b^{2} \leq a cos θ + b s in θ \leq a^{2} + b^{2}$
$∴ - 2 \leq cos θ \leq 2$
Hence max. value of $p + q$ is $2$

Q. If p and q are positive real numbers such that p2+q2=1, If the maximum value of (p+q) is equal to a​ then a is equal to

Given that p2+q2=1 ∴p=cosθ and q=sinθ Then p+q=cosθ+sinθ We know that −a2+b2​≤acosθ+bsinθ≤a2+b2​ ∴−2​≤cosθ≤2​ Hence max. value of p+q is 2​

Q. If $p$ and $q$ are positive real numbers such that $p^{2} + q^{2} = 1$ , If the maximum value of $(p + q)$ is equal to $a$ then a is equal to

Given that $p^{2} + q^{2} = 1$
$∴ p = cos θ$ and $q = s in θ$
Then $p + q = cos θ + s in θ$
We know that
$- a^{2} + b^{2} \leq a cos θ + b s in θ \leq a^{2} + b^{2}$
$∴ - 2 \leq cos θ \leq 2$
Hence max. value of $p + q$ is $2$