Question

If the line $y=1-4x$ touches the curve $y=x^3+a{x}^2-bx+c$ at $(0,1)$ and also touches the curv e $y=\frac{x^2}{2}-px+3$ 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 $\Rightarrow c=1$
$\text{ Also, }\frac{dy}{dx}=-4\Rightarrow -b=-4\Rightarrow b=4$
$\therefore b+c=5$
Also $1-4x=\frac{x^2}{2}-px+3\Rightarrow x^2+(8-2p)x+4=0$
$D=4(4-p{)}^2-16=0$
$(4-p)=\pm 2$
If$p=6\Rightarrow x=2$
$p=2\Rightarrow x=-2$
$\Rightarrow$ point of contact with second curve $(2,-7)$ on $(-2,9)$