Question

Let $R$ be the region in the first quadrant bounded by $x$ and $y$-axis and the graph of $f(x)=\frac{9}{25} x+b$ and $y=f^{-1}(x)$. If the area of $R$ is 49 , then the value of $b$ is

• A. $\frac{25}{9}$
• B. $\frac{28}{5}$
• C. $\frac{5}{28}$
• D. $\frac{9}{25}$

Answer 1

Answer: (B)

$f(x)=\frac{9}{25} x+b$
then $f ^{-1}( x )=\frac{25}{9}( x - b )$
science $f(x)$ and $f^{-1}(x)$ intersect at $y=x$.
Point of intersection is $\left(\frac{25}{16}( b ) \cdot \frac{25}{16} b \right)$
Required area $=2(\operatorname{ar} \Delta OAP ) \Rightarrow 2\left(\frac{1}{2} b \cdot \frac{25}{16} b \right)=49$
$ b ^2=\frac{49 \times 16}{25} $
$b =\frac{7 \times 4}{5}=\frac{28}{5}$