TD - Black-Scholes and a European forward-start call option

Question
In this question you must assume zero dividend yields, q = 0.
Show that V = αS, where α does not depend on S or t, is a solution of the
Black-Scholes equation.
Let C(S,t; T1, K) denote the Black-Scholes value of a European vanilla call
option with strike K and expiry T1 at time t and spot price S. Show that
the value of an “at the money” call is proportional to the spot (or strike).
A (European) forward-start call is a call option whose strike is not known at
the start of the contract but, rather, is agreed to be the spot price S0 a given
time T0 prior to expiry T1. The option can not be exercised prior to expiry
T1.
Show that the value v of a European forward-start call option is given by
V (S,t) =
(
αS if t ≤ T0,
C(S,t; T1, S0) if T0 ≤ t ≤ T1
where S0 is the spot price at time T0 and α is a function, which you should
determine, which depends only on T1 − T0, σ and r.
Briefly justify the ∆-hedging strategy implied by this price for t ≤ T0.