Question 1158181
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focus (0,6) and x-axis for the directrix means the parabola opens up, with vertex at (0,3).  The vertex form of the equation is<br>
{{{y-k = (1/(4p))(x-h)^2}}}<br>
where p is the directed distance from the vertex to the focus.<br>
So (h,k) = (0,3), and p=3.  The equation is<br>
{{{y-3 = (1/12)x^2}}}<br>
or<br>
{{{y = (1/12)x^2+3}}}<br>
When x=8, y=64/12+3 = 16/3+9/3 = 25/3.  So (8,25/3) is on the parabola.<br>
To find the equation of the tangent line, we have the coordinates of the point, so we need the slope.  Use calculus.<br>
y' = (1/6)x<br>
At x=8, the slope is 8/6 = 4/3.<br>
The equation of the tangent (point-slope form) is<br>
{{{y-25/3 = (4/3)(x-8)}}}<br>
Change to any equivalent form if required/desired.<br>