Mathematics - Application of Derivative Question with Solution | TestHub

$f^{\prime }(x)=2x-a$. At $(2,4),f^{\prime }(x)=4-a$. Equation of normal at $(2,4)$ is $(y-4)=\frac{-1}{(4-a)}(x-2)$. Let point of intersection with $x$ and $y$-axis be $A$ and $B$ respectively, then $A\equiv (-4a+18,0),B\equiv \left(0,\frac{4a-18}{a-4}\right)$. For the triangle to lie in the second quadrant, $a>\frac{9}{2}$. Area of triangle $=\frac{1}{2}(4a-18)\frac{(4a-18)}{(a-4)}=2\Rightarrow (4a-17)(a-5)=0\Rightarrow a=5$ or $\frac{17}{4}$. Only $a=5$ satisfies $a>\frac{9}{2}$.