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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.
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