Consider the hyperbola $B:\frac{(x-7)^2}{4}-\frac{(y+3)^2}{9}=1.$ A variable point $P(\alpha+7,\alpha^2-4)$ where $\alpha$ belongs to ${\Bbb R}$ exists in the $xy$-plane.
Let $B_{\rm left}$ and $B_{\rm right}$ be the left and right branches of the given hyperbola $B$.
Q1. The values of $\alpha$ for which two distinct real tangents can be drawn to $B_{\rm left}$ from $P$?
(Answer is $\alpha= -1$ [I think this is insufficient:
It should be an open interval containing $-1.$
--JM])
Q2. The values of $\alpha$ for which only one real tangent can be drawn to $B_{\rm left}$ only from point $P$?
(Answers are $\alpha= -2,-\frac12$)
Q3. The values of $\alpha$ for which two real tangents can be drawn to $B_{\rm right}$ only from point $P$?
(Answer is $\alpha=2$ [I think this is wrong:
It should be an open interval ending in but not containing $2.$
--JM])
These are the subparts which have to be solved from the data given above.
Is there any condition that should be satisfied in order to solve the question?
I am not able to form the inequalities to get the solution set for $\alpha.$
Hint: use geogebra to get intuition