Find an equation of the parabola described.
-Directrix the line y = 3; vertex at (0, 0)
A) $y = - \frac { 1 } { 12 } x ^ { 2 }$
B) $y = - 12 x ^ { 2 }$
C) $x = 3 y ^ { 2 }$
D) $x = - \frac { 1 } { 12 } y ^ { 2 }$

Question

Find an equation of the parabola described.
-Directrix the line y = 3; vertex at (0, 0) 
A) $y = - \frac { 1 } { 12 } x ^ { 2 }$
B) $y = - 12 x ^ { 2 }$
C) $x = 3 y ^ { 2 }$
D) $x = - \frac { 1 } { 12 } y ^ { 2 }$

Quizplus · Accepted Answer

The vertex form of a parabola is $y = a(x - h)^2 + k$, where (h, k) is the vertex. Given the vertex at (0, 0), the equation simplifies to $y = ax^2$. The directrix is y = 3, which is above the vertex, indicating the parabola opens downwards. The focus-directrix property of a parabola states that the distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix. Since the vertex is at (0, 0) and the directrix is y = 3, the focus is 3 units away from the directrix in the opposite direction, at y = -3. This means the parabola's focal length (p) is -3, indicating a = -1/(4p) = -1/(4*(-3)) = 1/12. Therefore, the equation is $y = -\frac{1}{12}x^2$.

Find an Equation of the Parabola Described y=−112x2y = - \frac { 1 } { 12 } x ^ { 2 }y=−121​x2

Find an Equation of the Parabola Described $y = - \frac { 1 } { 12 } x ^ { 2 }$