Question
Identify the focus and directrix of the parabola whose equation is (y-3)^2 = -4(x-5)
a) (3, -5)
b) (5, -3)
c) (5, 3)
d) (-5, 3)
a) (3, -5)
b) (5, -3)
c) (5, 3)
d) (-5, 3)
Answers
GPT-4o mini
The given equation of the parabola is \((y - 3)^2 = -4(x - 5)\). This is in the standard form \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex.
1. Identify the vertex:
- From the equation \((y - 3)^2 = -4(x - 5)\), we can see that:
- \(h = 5\)
- \(k = 3\)
- So, the vertex is at \((5, 3)\).
2. Determine \(p\):
- The coefficient \(-4\) indicates that \(4p = -4\), hence \(p = -1\). The negative sign indicates that the parabola opens to the left.
3. Identify the focus:
- The focus of a parabola that opens to the left is located \(p\) units to the left of the vertex. Therefore, we calculate the coordinates of the focus:
- Focus = \((h + p, k) = (5 - 1, 3) = (4, 3)\).
4. Identify the directrix:
- The directrix of a parabola that opens to the left is a vertical line located \(p\) units to the right of the vertex. Therefore:
- Directrix = \(x = h + p\) which is \(x = 5 - (-1) = 5 + 1 = 6\).
Putting this all together:
- The focus of the parabola is \((4, 3)\).
- The directrix is the line \(x = 6\).
However, none of the answer choices given appear to match this. Upon reviewing the options, it seems they may have intended the question to locate either the focus or the directrix inaccurately.
From the options:
- a) (3, -5)
- b) (5, -3)
- c) (5, 3) (this is the vertex, not the focus)
- d) (-5, 3)
The correct interpretation of the vertex and focus isn't listed in the choices. If the aim was to check the vertex rather than just the focus, c) (5, 3) might be considered correct. Otherwise, it appears there is a misunderstanding or typographical error regarding the choices given in relation to the derived focus and directrix.
1. Identify the vertex:
- From the equation \((y - 3)^2 = -4(x - 5)\), we can see that:
- \(h = 5\)
- \(k = 3\)
- So, the vertex is at \((5, 3)\).
2. Determine \(p\):
- The coefficient \(-4\) indicates that \(4p = -4\), hence \(p = -1\). The negative sign indicates that the parabola opens to the left.
3. Identify the focus:
- The focus of a parabola that opens to the left is located \(p\) units to the left of the vertex. Therefore, we calculate the coordinates of the focus:
- Focus = \((h + p, k) = (5 - 1, 3) = (4, 3)\).
4. Identify the directrix:
- The directrix of a parabola that opens to the left is a vertical line located \(p\) units to the right of the vertex. Therefore:
- Directrix = \(x = h + p\) which is \(x = 5 - (-1) = 5 + 1 = 6\).
Putting this all together:
- The focus of the parabola is \((4, 3)\).
- The directrix is the line \(x = 6\).
However, none of the answer choices given appear to match this. Upon reviewing the options, it seems they may have intended the question to locate either the focus or the directrix inaccurately.
From the options:
- a) (3, -5)
- b) (5, -3)
- c) (5, 3) (this is the vertex, not the focus)
- d) (-5, 3)
The correct interpretation of the vertex and focus isn't listed in the choices. If the aim was to check the vertex rather than just the focus, c) (5, 3) might be considered correct. Otherwise, it appears there is a misunderstanding or typographical error regarding the choices given in relation to the derived focus and directrix.