By y= 3^(1/2)x
I suppose you mean
cause otherwise it wouldn't be a line equation.
Well, you just replace the expression for y into the circle equation, which gives
x^2 + 8x + (3 - 0.5x)^2 + a = 0
After a little bit of manipulation, you will find a 2nd degree polynomial equation in the form
A x^2 + B x + C = 0.
at which point you will - indeed - compute the discriminant D = B^2 - 4 A C, which will be an expression with variable a.
(e.g. : 14 a - 3.5)
- if D > 0 there are 2 real solutions for x - i.e. 2 intersection points between the line and the circle
- if D<0 there are no real solutions for x - i.e. the line does not 'meet' the circle
- if D=0 - 1 real solution - the line is tangent to the circle
So you just solve D=0 to find a.