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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.)

Ask by Kelly Chen. in the United States
Mar 16,2025

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

Solución

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: \[ (x+2)^2 = -12(y-7). \]

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Bonus Knowledge

To find the equation of the parabola, we first note that the focus \((-2, 4)\) and the directrix \(y = 10\) indicate that this is a vertical parabola that opens downward, since the focus is below the directrix. The vertex of the parabola is located halfway between the focus and the directrix. The vertex thus lies at the point \((-2, \frac{4 + 10}{2}) = (-2, 7)\). Using the standard form of the equation for a vertical parabola \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus (or directrix), we find \(p = 4 - 7 = -3\). Substituting \(h = -2\), \(k = 7\), and \(p = -3\) into the equation gives us: \[ (x + 2)^2 = -12(y - 7) \] So, the equation of the parabola is \((x + 2)^2 = -12(y - 7)\). Happy graphing!

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