Question

question what is the standard form of the equation of the parabola with the focus (1, -3) and the directrix x = 7? note that standard form is (y - k)^2 = 4p(x - h). provide your answer below:

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus $(1,-3)$ and the point on the directrix $x = 7$ that has the same $y$ - coordinate as the focus. The $x$ - coordinate of the vertex $h$ is $\frac{1 + 7}{2}=4$, and the $y$ - coordinate of the vertex $k=-3$. So the vertex is $(4,-3)$.

Step2: Calculate the value of $p$

The distance between the vertex $(4,-3)$ and the focus $(1,-3)$ is $p$. Using the distance formula for the $x$ - coordinates (since the $y$ - coordinates are the same), $p=4 - 1=3$. Since the focus is to the left of the directrix, $p=- 3$.

Step3: Write the equation of the parabola

Substitute $h = 4$, $k=-3$, and $p=-3$ into the standard form $(y - k)^2=4p(x - h)$. We get $(y+3)^2=4\times(-3)(x - 4)$, which simplifies to $(y + 3)^2=-12(x - 4)$.

Answer:

$(y + 3)^2=-12(x - 4)$