Question

If the line $y=1-4 x$ touches the curve $y=x^3+a x^2-b x+c$ at $(0,1)$ and also touches the curv e $y=\frac{x^2}{2}-p x+3$ at $(\alpha, \beta)$ then

• A. $b+c=5$.
• B. sum of all possible value of $p$ is 8 .
• C. sum of all possible values of $\alpha$ is 0 .
• D. sum of all possible values of $\beta$ is 2 .

Answer 1

Answer: (A), (B), (C), (D)

$(0,1)$ lies on first curve $\Rightarrow c =1$
$\text { Also, } \frac{d y}{d x}=-4 \Rightarrow-b=-4 \Rightarrow b=4 $
$\therefore b+c=5$
Also $ 1-4 x=\frac{x^2}{2}-p x+3 \Rightarrow x^2+(8-2 p) x+4=0$
$D=4(4-p)^2-16=0 $
$(4-p)= \pm 2$
If$p =6 \Rightarrow x =2 $
$p =2 \Rightarrow x =-2$
$\Rightarrow$ point of contact with second curve $(2,-7)$ on $(-2,9)$