Find the standard form of the equation of the parabola with a focus at (0, -3) and a directrix at y = 3.

(y-k) = a(x-h)², where a is the coefficient, h & k, the coordinates of the vertex.

We have: focus(0, -3) and a directrix at y = 3.
Since the direcyrix y=3, and the focus (0,-3) then this parabola opens downward that means a<0, Moreover:
We know that the vertex is exactly halfway between the the focus & the directrix
So the vertex (0,0) and the equation becomes:
y-0 = a(x-h)²  →→ y = ax².
Ton calculate a, remember the formula a =1/4c , where c= the value of the focus, OR c= -3, then a=1/(-12) 

And the final equation y = - (1/2).x²

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12