SOLUTION: Verify that the point A = (8, 25/3) lies on the parabola whose focus is (0, 6) and whose 3 directrix is the x-axis. Find an equation for the line that is tangent to the parabola a

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Question 1158181: Verify that the point A = (8, 25/3) lies on the parabola whose focus is (0, 6) and whose 3
directrix is the x-axis. Find an equation for the line that is tangent to the parabola at A.
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


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

y-k+=+%281%2F%284p%29%29%28x-h%29%5E2

where p is the directed distance from the vertex to the focus.

So (h,k) = (0,3), and p=3. The equation is

y-3+=+%281%2F12%29x%5E2

or

y+=+%281%2F12%29x%5E2%2B3

When x=8, y=64/12+3 = 16/3+9/3 = 25/3. So (8,25/3) is on the parabola.

To find the equation of the tangent line, we have the coordinates of the point, so we need the slope. Use calculus.

y' = (1/6)x

At x=8, the slope is 8/6 = 4/3.

The equation of the tangent (point-slope form) is

y-25%2F3+=+%284%2F3%29%28x-8%29

Change to any equivalent form if required/desired.