Question
Identify the focus and directrix of the parabola whose equation is (y-3)^2 = -4(x-5)
a) (4, 2)
b) (4, 4)
c) (4, 3)
d) (6, 3)
a) (4, 2)
b) (4, 4)
c) (4, 3)
d) (6, 3)
Answers
GPT-4o mini
The equation of the parabola given is \((y - 3)^2 = -4(x - 5)\).
This can be rewritten in the standard form of a parabola that opens to the left:
\[
(y - k)^2 = -4p(x - h)
\]
where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the focus (and also to the directrix).
From the equation \((y - 3)^2 = -4(x - 5)\), we can identify:
- The vertex \((h, k) = (5, 3)\)
- The value of \(4p = 4\), therefore \(p = 1\).
Since the parabola opens to the left (because of the negative sign), the focus is located \(p\) units to the left of the vertex, and the directrix is \(p\) units to the right of the vertex.
Calculating the focus:
- Focus: \((h - p, k) = (5 - 1, 3) = (4, 3)\)
Calculating the directrix:
- Directrix: \(x = h + p = 5 + 1 = 6\)
So, the focus is at \((4, 3)\) and the directrix is the line \(x = 6\).
Thus, the correct answer for the focus of the parabola is:
**c) (4, 3)**
This can be rewritten in the standard form of a parabola that opens to the left:
\[
(y - k)^2 = -4p(x - h)
\]
where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the focus (and also to the directrix).
From the equation \((y - 3)^2 = -4(x - 5)\), we can identify:
- The vertex \((h, k) = (5, 3)\)
- The value of \(4p = 4\), therefore \(p = 1\).
Since the parabola opens to the left (because of the negative sign), the focus is located \(p\) units to the left of the vertex, and the directrix is \(p\) units to the right of the vertex.
Calculating the focus:
- Focus: \((h - p, k) = (5 - 1, 3) = (4, 3)\)
Calculating the directrix:
- Directrix: \(x = h + p = 5 + 1 = 6\)
So, the focus is at \((4, 3)\) and the directrix is the line \(x = 6\).
Thus, the correct answer for the focus of the parabola is:
**c) (4, 3)**