Verified answer

First find where the chord cuts the parabola by solving their equations simultaneously.

5x - x^2 + 36 = 0 ----> (9 - x)(4 + x) = 0 ----> x = 9 or -5.

Calculate y values from either equation.

x = 9 ----> y = 9*9/12 = 27/4

x = -5 ----> y = -5*-5/12 = 25/12

Now find gradient of tangent at each of these points.

y = (x^2)/12 ----> dy/dx = x/6

At (9, 27/4) gradient = 9/6 = 3/2

Equation of tangent y - 27/4 = (3/2)(x - 9) ----> y = 3x/2 -27/4

At (-5, 25/12) gradient = -5/6

Equation of tangent y - 25/12 = (-5/6)(x + 5) ----> y = -5x/6 - 25/12

These tangents cross when

3x/2 - 27/4 = -5x/6 - 25/12 ----> 18x - 81 = -10x - 25

----> 28x = 56 ----> x = 2

Find y value from either tangent equation

y = 3*2/2 - 27/4 = 3 - 27/4 = -15/4

or

y = -5*2/6 - 25/12 = -20/12 - 25/12 = -45/12 = -15/4

The tangents intersect at (2, -15/4)

EDIT. Yes, I can't believe what I wrote in my second line. Definitely a senior moment!

Reworking with x = -4 confirms the result below.

Mathman is exactly right, except for the first part, where x= -4 or 9 ( from the factors)

This uses calculus, but there is also an old geometry formula from Euclid.

If two tangents of a parabola intersect at an external point, then their intersection lies on the directrix, and the other coordinate is the average of the endpoints.

So you found where the parabola intersects the chord. (-4,4/3) and (9,27/4)

The p value for the parabola is 3, so the directrix is y= -3

so average the x coordinates .x = ( -4+9)/2= 5/2 and y= -3

In other words if P and Q are the endpoints of the chord, and you drop perpendiculars from P and Q to the directrix, at P' and Q', then the intersection point is the midpoint of P'Q'

Hoping this helps!