Maths question help

Hi, please could I have help on this question? I’m confused as to why in the ms, they found the derivative and equated it to zero, I’m unsure how to approach this question?
Thanks!

Sorry, meant to say it’s part iii I need help with

the shortest distance also corresponds to the minimum d^2. Its probably a bit simpler to note its a hidden quadratic so complete the square / write down the minimum of d^2 as
c - b^2/(4a) = 2 - 100/52 = 1/13
so dmin is the root of that. Obv you could differentiate and set equal to zero as that corresponds to the minimum distance.

ohhh ok i get what you are saying about diffrentiating and equalling it to 0 but I'm confused about why you have used the discriminant? the discriminant is b^2-4ac but I'm getting a negative value, (-10)^2-4(13)(2) which is-4? I don't get what we are supposed to do from here? I identified it as a hidden quadratic but couldn't solve it as it gave imaginary numbers

Im not using the disciminant, though it is related. The discriminant "must" be negative for the hidden quadratic as d^2>=0 so the hidden quadratic cannot cross the x-axis so there are no (at most 1) real roots.
But a quadratic in completed square form is
ax^2 + bx + c = a(x+b/2a)^2 + c-b^2/4a
The c-b^2/(4a) is the min/max value of the quadratic at x=-b/(2a).
Putting the small amount of extra analysis to put it in the usual quadratic equation/roots form shows how its transformed slightly to the discriminant.