Find expert answers and community support for all your questions on IDNLearn.com. Our experts are ready to provide in-depth answers and practical solutions to any questions you may have.
Sagot :
To determine the coordinates of the focus for the satellite dish described by the equation [tex]\(x^2 = 8y\)[/tex], we need to recognize that this is a parabolic equation in standard form. For a parabola that opens upwards and has its vertex at the origin [tex]\((0,0)\)[/tex], the standard equation is given by:
[tex]\[ x^2 = 4py \][/tex]
Where [tex]\((0, p)\)[/tex] is the focus of the parabola.
1. Identify the given equation and compare it with the standard form:
The given equation is:
[tex]\[ x^2 = 8y \][/tex]
2. Match the given equation to the standard form [tex]\(x^2 = 4py\)[/tex]:
[tex]\[ x^2 = 4py \][/tex]
By comparing it with:
[tex]\[ x^2 = 8y \][/tex]
We see that [tex]\(4p = 8\)[/tex].
3. Solve for [tex]\(p\)[/tex]:
[tex]\[ 4p = 8 \][/tex]
[tex]\[ p = \frac{8}{4} \][/tex]
[tex]\[ p = 2 \][/tex]
4. Determine the coordinates of the focus:
Since the focus is located at [tex]\((0, p)\)[/tex], we substitute [tex]\(p = 2\)[/tex] into the coordinates.
Thus, the coordinates of the focus are [tex]\((0, 2)\)[/tex].
Therefore, the coordinates of the focus of the satellite dish described by the equation [tex]\(x^2 = 8y\)[/tex] are:
[tex]\[ \boxed{(0, 2)} \][/tex]
[tex]\[ x^2 = 4py \][/tex]
Where [tex]\((0, p)\)[/tex] is the focus of the parabola.
1. Identify the given equation and compare it with the standard form:
The given equation is:
[tex]\[ x^2 = 8y \][/tex]
2. Match the given equation to the standard form [tex]\(x^2 = 4py\)[/tex]:
[tex]\[ x^2 = 4py \][/tex]
By comparing it with:
[tex]\[ x^2 = 8y \][/tex]
We see that [tex]\(4p = 8\)[/tex].
3. Solve for [tex]\(p\)[/tex]:
[tex]\[ 4p = 8 \][/tex]
[tex]\[ p = \frac{8}{4} \][/tex]
[tex]\[ p = 2 \][/tex]
4. Determine the coordinates of the focus:
Since the focus is located at [tex]\((0, p)\)[/tex], we substitute [tex]\(p = 2\)[/tex] into the coordinates.
Thus, the coordinates of the focus are [tex]\((0, 2)\)[/tex].
Therefore, the coordinates of the focus of the satellite dish described by the equation [tex]\(x^2 = 8y\)[/tex] are:
[tex]\[ \boxed{(0, 2)} \][/tex]
Your engagement is important to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.