At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine the required values, let’s break it down step by step.
### Finding the Radius of the Circle
First, we need to calculate the radius of the circle. The radius is the distance from the center of the circle, which is at [tex]\((-7, -1)\)[/tex], to any point on the circle. We are given that the circle passes through the point [tex]\((8, 7)\)[/tex].
To find the distance (radius), we use the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the coordinates of the center and the given point:
[tex]\[ \text{Radius} = \sqrt{(8 - (-7))^2 + (7 - (-1))^2} = \sqrt{(8 + 7)^2 + (7 + 1)^2} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \][/tex]
So, the radius of the circle is [tex]\(\boxed{17}\)[/tex] units.
### Finding the y-coordinate of the Point [tex]\((-15, y)\)[/tex] on the Circle
Now, we need to find the [tex]\(y\)[/tex]-coordinate for the point [tex]\((-15, y)\)[/tex] that lies on the circle. The equation of the circle centered at [tex]\((-7, -1)\)[/tex] with radius 17 is:
[tex]\[ (x + 7)^2 + (y + 1)^2 = 17^2 \][/tex]
Substituting [tex]\(x = -15\)[/tex]:
[tex]\[ (-15 + 7)^2 + (y + 1)^2 = 17^2 \][/tex]
Simplifying inside the parentheses:
[tex]\[ (-8)^2 + (y + 1)^2 = 289 \][/tex]
[tex]\[ 64 + (y + 1)^2 = 289 \][/tex]
Rearranging to solve for [tex]\((y + 1)^2\)[/tex]:
[tex]\[ (y + 1)^2 = 289 - 64 \][/tex]
[tex]\[ (y + 1)^2 = 225 \][/tex]
Taking the square root of both sides:
[tex]\[ y + 1 = \pm 15 \][/tex]
This gives us two possible solutions:
[tex]\[ y + 1 = 15 \quad \text{or} \quad y + 1 = -15 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = 14 \quad \text{or} \quad y = -16 \][/tex]
Thus, the two possible points [tex]\((-15, y)\)[/tex] on the circle are [tex]\((-15, 14)\)[/tex] and [tex]\((-15, -16)\)[/tex].
Since we need only one value for the point [tex]\((-15, y)\)[/tex] in the given problem, we can write the final answer using either of the [tex]\(y\)[/tex]-coordinates. One possible answer would be:
[tex]\[ (-15, 14) \][/tex]
Therefore, the correct values are:
- The radius of the circle is [tex]\(\boxed{17}\)[/tex] units.
- The point [tex]\((-15, 14)\)[/tex] lies on the circle, so the second box should be [tex]\(\boxed{14}\)[/tex]. Alternatively, if considering the other possible [tex]\(y\)[/tex]-coordinate, the second box could also be [tex]\(\boxed{-16}\)[/tex].
Either [tex]\(\boxed{14}\)[/tex] or [tex]\(\boxed{-16}\)[/tex] is an acceptable answer for the second blank, depending on which specific point you consider.
### Finding the Radius of the Circle
First, we need to calculate the radius of the circle. The radius is the distance from the center of the circle, which is at [tex]\((-7, -1)\)[/tex], to any point on the circle. We are given that the circle passes through the point [tex]\((8, 7)\)[/tex].
To find the distance (radius), we use the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the coordinates of the center and the given point:
[tex]\[ \text{Radius} = \sqrt{(8 - (-7))^2 + (7 - (-1))^2} = \sqrt{(8 + 7)^2 + (7 + 1)^2} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \][/tex]
So, the radius of the circle is [tex]\(\boxed{17}\)[/tex] units.
### Finding the y-coordinate of the Point [tex]\((-15, y)\)[/tex] on the Circle
Now, we need to find the [tex]\(y\)[/tex]-coordinate for the point [tex]\((-15, y)\)[/tex] that lies on the circle. The equation of the circle centered at [tex]\((-7, -1)\)[/tex] with radius 17 is:
[tex]\[ (x + 7)^2 + (y + 1)^2 = 17^2 \][/tex]
Substituting [tex]\(x = -15\)[/tex]:
[tex]\[ (-15 + 7)^2 + (y + 1)^2 = 17^2 \][/tex]
Simplifying inside the parentheses:
[tex]\[ (-8)^2 + (y + 1)^2 = 289 \][/tex]
[tex]\[ 64 + (y + 1)^2 = 289 \][/tex]
Rearranging to solve for [tex]\((y + 1)^2\)[/tex]:
[tex]\[ (y + 1)^2 = 289 - 64 \][/tex]
[tex]\[ (y + 1)^2 = 225 \][/tex]
Taking the square root of both sides:
[tex]\[ y + 1 = \pm 15 \][/tex]
This gives us two possible solutions:
[tex]\[ y + 1 = 15 \quad \text{or} \quad y + 1 = -15 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = 14 \quad \text{or} \quad y = -16 \][/tex]
Thus, the two possible points [tex]\((-15, y)\)[/tex] on the circle are [tex]\((-15, 14)\)[/tex] and [tex]\((-15, -16)\)[/tex].
Since we need only one value for the point [tex]\((-15, y)\)[/tex] in the given problem, we can write the final answer using either of the [tex]\(y\)[/tex]-coordinates. One possible answer would be:
[tex]\[ (-15, 14) \][/tex]
Therefore, the correct values are:
- The radius of the circle is [tex]\(\boxed{17}\)[/tex] units.
- The point [tex]\((-15, 14)\)[/tex] lies on the circle, so the second box should be [tex]\(\boxed{14}\)[/tex]. Alternatively, if considering the other possible [tex]\(y\)[/tex]-coordinate, the second box could also be [tex]\(\boxed{-16}\)[/tex].
Either [tex]\(\boxed{14}\)[/tex] or [tex]\(\boxed{-16}\)[/tex] is an acceptable answer for the second blank, depending on which specific point you consider.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
