Question

The polynomial $P\left(x\right)=x^3+a{x}^2+bx+c$ has the property that the mean of it's If the zeroes, and the sum of its coefficients are all equal. If the $y$ intercept of the graph of $y=P\left(x\right)$ is $2$, then the value of $b$ is

• A. $-11$
• B. $-9$
• C. $7$
• D. $5$

Answer 1

Answer: (C)

$7$

The center of the hyperbola $O$ midpoint of the line going two focii

So the co ordinates of the center are

$(\frac{8+0}{2},\frac{3+3}{2})i.e,(4,3)$

Let $2a$ and $2b$ be the length of the transverse and conjugate axis and let e be accentricity equation of hyperbola

$\frac{(x-4{)}^2}{a^2}-\frac{(y-3{)}^2}{b^2}=1$

Distance between the two focii $=2ae$

$\sqrt{(8-0{)}^2+(3-3{)}^2}=2aeae=4a=3b^2=a^2(e^2-1^2)\because b^2=9(-1+\frac{16}{9})=7$

Equation of hyperbola

$\frac{(x-4{)}^2}{9}+\frac{(y-3{)}^2}{7}=17x^2-9y^2-56x+54y-32=0$