Question

Let $a, b, c \in R$ be such that $2 a+3 b+6 c=0$ and $g(x)$ be the anti derivative of $f(x)=a x^{2}+b x +c .$ If the slopes of the tangents drawn to the curve $y=g(x)$ at $(1, g( l ))$ and $(2, g(2))$ are equal, then

• A. $\frac{a}{3}=\frac{b}{-8}=\frac{c}{3}$
• B. $\frac{a}{6}=\frac{b}{-18}=\frac{c}{7}$
• C. $\frac{a}{3}=\frac{b}{-6}=\frac{c}{2}$
• D. $a=b=c=-1$

Answer 1

Answer: (B)

We have,
$2 a + 3 b + 6 c=0$ ...(i)
$g(x)$ is anti derivative of $f(x)=a x^{2}+b x+ c$
$\therefore g'(x)=a x^{2}+ b x +c$
Given, $g'(1) =g'(2)$
$\Rightarrow a+ b +c =4 a+2 b +c$
$\Rightarrow 3 a +b =0$ ...(ii)
From Eqs. (i) and (ii), we get
$\frac{a}{-6}=\frac{b}{18}=\frac{c}{-7}$
$\Rightarrow \frac{a}{6}=\frac{b}{-18}=\frac{c}{7}$