Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine which of the given parabolic equations have the directrix \(x = -4\), we need to use the following properties of a parabola in the form:
[tex]\[ x = \frac{1}{4a} y^2 + b y + c \][/tex]
The directrix for this parabola is given by:
[tex]\[ x = h - \frac{1}{4a} \][/tex]
where \( h \) is the x-coordinate of the vertex. Setting the directrix to \( x = -4 \), we get:
[tex]\[ -4 = h - \frac{1}{4a} \][/tex]
[tex]\[ h = -4 + \frac{1}{4a} \][/tex]
We will check each equation to see if their vertex \( h = -4 \).
1. Equation: \( x = \frac{y^2}{24} - \frac{7y}{12} + \frac{97}{24} \)
- Here, \( a = \frac{1}{24} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{24}} = 6\)
- Thus, \( h = -4 + 6 = 2 \neq -4 \)
2. Equation: \( x = -\frac{y^2}{16} + \frac{5y}{8} - \frac{153}{16} \)
- Here, \( a = -\frac{1}{16} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \left(-\frac{1}{16}\right)} = -4\)
- Thus, \( h = -4 + (-4) = -8 \neq -4 \)
3. Equation: \( x = -\frac{y^2}{12} + \frac{y}{2} - \frac{39}{4} \)
- Here, \( a = -\frac{1}{12} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \left(-\frac{1}{12}\right)} = -3\)
- Thus, \( h = -4 + (-3) = -7 \neq -4 \)
4. Equation: \( x = -\frac{y^2}{28} - \frac{5y}{7} - \frac{95}{7} \)
- Here, \( a = -\frac{1}{28} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \left(-\frac{1}{28}\right)} = -7\)
- Thus, \( h = -4 + (-7) = -11 \neq -4 \)
5. Equation: \( x = \frac{y^2}{48} + \frac{5y}{24} + \frac{58}{48} \)
- Here, \( a = \frac{1}{48} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{48}} = 12\)
- Thus, \( h = -4 + 12 = 8 \neq -4 \)
6. Equation: \( x = \frac{y^2}{32} + \frac{3y}{16} + \frac{137}{32} \)
- Here, \( a = \frac{1}{32} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{32}} = 8\)
- Thus, \( h = -4 + 8 = 4 \neq -4 \)
Therefore, none of the given equations have the directrix [tex]\(x = -4\)[/tex].
[tex]\[ x = \frac{1}{4a} y^2 + b y + c \][/tex]
The directrix for this parabola is given by:
[tex]\[ x = h - \frac{1}{4a} \][/tex]
where \( h \) is the x-coordinate of the vertex. Setting the directrix to \( x = -4 \), we get:
[tex]\[ -4 = h - \frac{1}{4a} \][/tex]
[tex]\[ h = -4 + \frac{1}{4a} \][/tex]
We will check each equation to see if their vertex \( h = -4 \).
1. Equation: \( x = \frac{y^2}{24} - \frac{7y}{12} + \frac{97}{24} \)
- Here, \( a = \frac{1}{24} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{24}} = 6\)
- Thus, \( h = -4 + 6 = 2 \neq -4 \)
2. Equation: \( x = -\frac{y^2}{16} + \frac{5y}{8} - \frac{153}{16} \)
- Here, \( a = -\frac{1}{16} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \left(-\frac{1}{16}\right)} = -4\)
- Thus, \( h = -4 + (-4) = -8 \neq -4 \)
3. Equation: \( x = -\frac{y^2}{12} + \frac{y}{2} - \frac{39}{4} \)
- Here, \( a = -\frac{1}{12} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \left(-\frac{1}{12}\right)} = -3\)
- Thus, \( h = -4 + (-3) = -7 \neq -4 \)
4. Equation: \( x = -\frac{y^2}{28} - \frac{5y}{7} - \frac{95}{7} \)
- Here, \( a = -\frac{1}{28} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \left(-\frac{1}{28}\right)} = -7\)
- Thus, \( h = -4 + (-7) = -11 \neq -4 \)
5. Equation: \( x = \frac{y^2}{48} + \frac{5y}{24} + \frac{58}{48} \)
- Here, \( a = \frac{1}{48} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{48}} = 12\)
- Thus, \( h = -4 + 12 = 8 \neq -4 \)
6. Equation: \( x = \frac{y^2}{32} + \frac{3y}{16} + \frac{137}{32} \)
- Here, \( a = \frac{1}{32} \)
- \(\frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{32}} = 8\)
- Thus, \( h = -4 + 8 = 4 \neq -4 \)
Therefore, none of the given equations have the directrix [tex]\(x = -4\)[/tex].
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.