Find the focus and directrix of the following parabola: $$ (y-6)^{2}=12(x-2) $$ Focus: $$ ([?], \square]) $$ Directrix: $$ x=\square $$

Question

UpStudy AI · Accepted Answer

Given the parabola
$$
(y-6)^2=12(x-2)
$$
we compare with the standard form
$$
(y-k)^2=4p(x-h).
$$
**Step 1:** Identify $$h$$ and $$k$$.

From the given equation, we have:
$$
h = 2 \quad \text{and} \quad k = 6.
$$

**Step 2:** Determine $$p$$.

Since the equation is in the form $$(y-6)^2=12(x-2)$$ and $$4p=12$$, we solve for $$p$$:
$$
4p = 12 \quad \Rightarrow \quad p = 3.
$$

**Step 3:** Find the focus.

For a parabola that opens to the right, the focus is located at:
$$
(h+p, k).
$$
Thus, the focus is:
$$
(2+3, 6)=(5,6).
$$

**Step 4:** Find the directrix.

For a parabola that opens to the right, the directrix is the vertical line:
$$
x = h-p.
$$
So the directrix is:
$$
x = 2-3 = -1.
$$

**Final Answer:**

Focus: $$(5,6)$$

Directrix: $$x = -1$$
Focus: (5, 6) Directrix: x = -1

Find the focus and directrix of the following parabola: \( (y-6)^{2}=12(x-2) \) Focus: \( ([?], \square]) \) Directrix: \( x=\square \)