Question

If $\forall h\in R-\left\{0\right\},$ two distinct tangents can be drawn from the points $(2+h,3h-1)$ to the curve $y=x^3-6x^2-a+bx$, then $\frac{a}{b}$ is

Answer 1

One of the tangents must pass through the the point of inflection of the curve.
$y^{\prime \prime }=0\Rightarrow x=2$
$P(2,2b-a-16)$
Equation of tangent at P is $y=(b-12)x-a+8$ ...(1)
Let Q = (2 + h, 3h - 1)
Locus of Q $\to 3x-y=7$ ...(2)
(1) and (2) must be identical.
a = 15, b = 15