Answer:
Part 1) The radius is ![r=17\ units](https://tex.z-dn.net/?f=r%3D17%5C%20units)
Part 2) The points (-15,14) and (-15,-16) lies on the circle
Step-by-step explanation:
<u><em>The correct question is</em></u>
A circle is centered at the point (-7, -1) and passes through the point (8, 7).
The radius of the circle is ____units.
The point (-15,
?) ) lies on this circle
step 1
Find the radius of the circle
we know that
The distance from the center to any point on the circumference is equal to the radius.
we have
![(-7,-1),(8,7)](https://tex.z-dn.net/?f=%28-7%2C-1%29%2C%288%2C7%29)
Find the distance
the formula to calculate the distance between two points is equal to
![d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28y2-y1%29%5E%7B2%7D%2B%28x2-x1%29%5E%7B2%7D%7D)
substitute
![r=\sqrt{(7+1)^{2}+(8+7)^{2}}\\r=\sqrt{(8)^{2}+(15)^{2}}\\r=\sqrt{289}\\r=17\ units](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%287%2B1%29%5E%7B2%7D%2B%288%2B7%29%5E%7B2%7D%7D%5C%5Cr%3D%5Csqrt%7B%288%29%5E%7B2%7D%2B%2815%29%5E%7B2%7D%7D%5C%5Cr%3D%5Csqrt%7B289%7D%5C%5Cr%3D17%5C%20units)
step 2
The point (-15,y) lies on the circle
we know that
If a ordered pair lie on the circle, them the ordered pair must satisfy the equation of the circle
The equation of the circle is
![(x-h)^{2}+(y-k)^{2}=r^{2}](https://tex.z-dn.net/?f=%28x-h%29%5E%7B2%7D%2B%28y-k%29%5E%7B2%7D%3Dr%5E%7B2%7D)
The center is (-7,-1) an the radius is 17 units
substitute
![(x+7)^{2}+(y+1)^{2}=17^{2}\\(x+7)^{2}+(y+1)^{2}=289](https://tex.z-dn.net/?f=%28x%2B7%29%5E%7B2%7D%2B%28y%2B1%29%5E%7B2%7D%3D17%5E%7B2%7D%5C%5C%28x%2B7%29%5E%7B2%7D%2B%28y%2B1%29%5E%7B2%7D%3D289)
substitute the x-coordinate of the point and solve for the y-coordinate
For x=-15
![(-15+7)^{2}+(y+1)^{2}=289\\(-8)^{2}+(y+1)^{2}=289\\64+(y+1)^{2}=289\\(y+1)^{2}=289-64\\(y+1)^{2}=225](https://tex.z-dn.net/?f=%28-15%2B7%29%5E%7B2%7D%2B%28y%2B1%29%5E%7B2%7D%3D289%5C%5C%28-8%29%5E%7B2%7D%2B%28y%2B1%29%5E%7B2%7D%3D289%5C%5C64%2B%28y%2B1%29%5E%7B2%7D%3D289%5C%5C%28y%2B1%29%5E%7B2%7D%3D289-64%5C%5C%28y%2B1%29%5E%7B2%7D%3D225)
square root both sides
![(y+1)=\pm15\\y=-1\pm15\\y=-1(+)15=14\\y=-1(-)15=-16](https://tex.z-dn.net/?f=%28y%2B1%29%3D%5Cpm15%5C%5Cy%3D-1%5Cpm15%5C%5Cy%3D-1%28%2B%2915%3D14%5C%5Cy%3D-1%28-%2915%3D-16)
therefore
The points (-15,14) and (-15,-16) lies on the circle
see the attached figure to better understand the problem