Name Cumulative Test Due Monday $$ 3 / 17 / 25 $$ 10. (4pt) Determine an equation for the parabola with focus. $$ (-2,4) $$ and directrix $$ y=10 $$. (The use of the grid below is optional.)

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1. The definition of a parabola is the set of all points $$(x,y)$$ that are equidistant from the focus $$(-2,4)$$ and the directrix $$y=10$$. Thus, for any point $$(x,y)$$ on the parabola,  
   $$
   \sqrt{(x+2)^2+(y-4)^2} = |y-10|.
   $$

2. Square both sides to eliminate the square root:  
   $$
   (x+2)^2+(y-4)^2 = (y-10)^2.
   $$

3. Expand each term:  
   $$
   \begin{aligned}
   (x+2)^2 &= x^2+4x+4, \
   (y-4)^2 &= y^2-8y+16, \
   (y-10)^2 &= y^2-20y+100.
   \end{aligned}
   $$
   Substituting these gives:  
   $$
   x^2+4x+4+y^2-8y+16 = y^2-20y+100.
   $$

4. Simplify the equation by canceling $$ y^2 $$ from both sides and combining like terms:  
   $$
   x^2+4x+4-8y+16 = -20y+100.
   $$
   Combine the constants $$4+16$$ on the left side:  
   $$
   x^2+4x+20-8y = -20y+100.
   $$

5. Isolate $$ y $$ terms on one side:  
   $$
   x^2+4x+20-8y+20y-100 = 0,
   $$
   which simplifies to:  
   $$
   x^2+4x+12y-80 = 0.
   $$

6. An alternative approach is to find the vertex of the parabola. The vertex lies midway between the focus and the directrix.  
   - The $$y$$-coordinate of the vertex is  
     $$
     k=\frac{4+10}{2}=7.
     $$
   - Since the focus has $$x=-2$$, the vertex has $$h=-2$$.  
     
   The distance from the vertex to the focus is  
   $$
   p=7-4=3,
   $$
   and since the focus is below the vertex, we have $$p=-3$$.  
     
   In vertex form for a vertical parabola,  
   $$
   (x-h)^2=4p(y-k),
   $$
   substituting $$h=-2$$, $$k=7$$, and $$p=-3$$ gives:  
   $$
   (x+2)^2=4(-3)(y-7)= -12(y-7).
   $$

7. The equation of the parabola is therefore:  
   $$
   (x+2)^2 = -12(y-7).
   $$
The equation of the parabola is $$ (x + 2)^2 = -12(y - 7) $$.

Name Cumulative Test Due Monday \( 3 / 17 / 25 \) 10. (4pt) Determine an equation for the parabola with focus. \( (-2,4) \) and directrix \( y=10 \). (The use of the grid below is optional.)

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