Hi there! Thank you for the opportunity to help you with your question!
The first part for solving this is rewriting this equation in vertex form y = a(x-h)^2 + k
8y = -x^2 +6x + 7
y = -(x^2-6x-7)/8
Then we have to complete the square:
y=-[(x-3)^2 - 9 - 7]/8 = -[(x-3)^2]/8 + 2
Going back to standard form, this says a = -1/8, h = 3 and k =2.
1. When the equation is in vertex form, the focus is at the point (h, k + 1/(4a) ). Replacing h, k, and a we get that the focus is at (3, 2-1/(4/8) ) = (3, 0)
Step 2. The directrix of the parabola is given by the line y = k -1/(4a) = 2 + 2 = 4
y = 4 is the equation of the directrix
The vertex is located at (h,k), or (3, 2)
The points which lie on the line through the focus and parallel to the directrix are the points that have y = 0
0 = -(x^2-6x-7)/8
(x-7)(x+1) = 0
x = 7, -1
So the points are (7,0), (-1,0)Please let me know if you need any clarification. Always glad to help!
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