Question

question 4 (5 points) find the equation of the parabola with its focus at (-7,6) and its directrix at y = 4. a) y = 1/4(x - 7)^2+5 b) y = 1/4(x + 7)^2+5 c) y = 1/4(x + 5)^2+7 d) y = 1/8(x + 7)^2+5

Explanation:

Step1: Find the vertex of the parabola

The vertex of a parabola is the mid - point between the focus and the point on the directrix directly below (or above) the focus. The x - coordinate of the vertex is the same as the x - coordinate of the focus, $x=-7$. The y - coordinate of the vertex is the mid - point between the y - coordinate of the focus and the y - value of the directrix. So, $y=\frac{6 + 4}{2}=5$. The vertex is $(-7,5)$.

Step2: Determine the value of $p$

The distance $p$ between the vertex and the focus (or the vertex and the directrix) is $|6 - 5| = 1$.

Step3: Write the equation of the parabola

The standard form of a parabola with a vertical axis of symmetry is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex and $a=\frac{1}{4p}$. Since $p = 1$, $a=\frac{1}{4}$, and $(h,k)=(-7,5)$. The equation of the parabola is $y=\frac{1}{4}(x + 7)^2+5$.

Answer:

B. $y=\frac{1}{4}(x + 7)^2+5$