Question

Let $A\left(k\right)$ be the area bounded by the curves $y=x^2-3$ and $y=kx+2.$ Then

• A. the range of $A\left(k\right)$ is $[\frac{10\sqrt{5}}{3},\infty )$
• B. the range of $A\left(k\right)$ is $[\frac{20\sqrt{5}}{3},\infty )$
• C. If function $k\to A\left(k\right)$ is defined for $k\in [-2,\infty ),$ then $A\left(k\right)$ is many- one function
• D. the value of $k$ for which area is minimum is $1$

Answer 1

Answer: (B), (C)

The correct options are
B the range of $A\left(k\right)$ is $[\frac{20\sqrt{5}}{3},\infty )$
C If function $k\to A\left(k\right)$ is defined for $k\in [-2,\infty ),$ then $A\left(k\right)$ is many- one function


Line $y=kx+2$ passes through fixed point $(0,2)$ for different value of $k$.

Also, the minimum $A\left(k\right)$ occurs when $k=0$, as when line is rotated from this position about point $(0,2)$, the increased part of area is more than the decreased part of area.

$\therefore A(k{)}_{\text{min}}=2\underset{0}{\overset{\sqrt{5}}{\int }}[2-(x^2-3)]dx$
$=2{[5x-\frac{x^3}{3}]}_0^{\sqrt{5}}$
$=\frac{20\sqrt{5}}{3}\text{sq. units}$