1.27: Past-paper question 8

The curve $\displaystyle C$ has equation $\displaystyle y = f(x)$

The line $\displaystyle l$ with equation $\displaystyle y - 8x + 9 = 0$ is a tangent to $\displaystyle C$

The line $\displaystyle k$ is perpendicular to $\displaystyle l$ and is also the tangent to $\displaystyle C$ at the point where $\displaystyle x = p$

(a) Find the value of $\displaystyle p$

(b) Show that

The quadratic equation $\displaystyle f(x) = 0$ has roots $\displaystyle \alpha$ and $\displaystyle \beta$

Without solving the equation and using your results from part (b)

(c) form a quadratic equation with integer coefficients, that has roots