In the coordinate plane, the region M consists of all points ( x , y ) satisfying the inequalities y ≥ 0 , y ≤ x , and y ≤ 2 − x simultaneously. The region N which varies with parameter t , consists o

Question

In the coordinate plane, the region M consists of all points ( $x, y$ ) satisfying the inequalities $y \geq 0, y \leq x$ , and $y \leq 2 - x$ simultaneously. The region N which varies with parameter t , consists of all the points $(x, y)$ satisfying the inequalities $t \leq x \leq t + 1$ and $0 \leq t \leq 1$ simultaneously. If the area of the region $M \cap N$ is a function of t , i,e., $M \cap N = f (t)$ and if $α$ is the value of t for which this area is maximum, then the numerical value of $2 α$ is

Answer:

1

Solution:

$M \cap N = f (t) = - t^{2} + t + 1/2$

$= \frac{3}{4} - (t - \frac{1}{2})^{2} f (t)$ is maximum for $t = 1/2$ i.e. $α = \frac{1}{2} \Rightarrow 2 α = 1$

TestHub · Accepted Answer

M ∩ N = f ( t ) = − t 2 + t + 1/2 = 4 3 ​ − ( t − 2 1 ​ ) 2 f ( t ) is maximum for t = 1/2 i.e. α = 2 1 ​ ⇒ 2 α = 1

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