Answer:
Part 1) The radius of the circle is [tex]r=17\ units[/tex]
Part 2) The point (-15,14) and the point (-15,-16) lies on the circle
Step-by-step explanation:
step 1
Find the radius of the circle
we know that
To find the radius of the circle calculate the distance between the center of the circle and the point (8,7)
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
[tex](-7,-1)\\(8,7)[/tex]
substitute
[tex]r=\sqrt{(7+1)^{2}+(8+7)^{2}}[/tex]
[tex]r=\sqrt{(8)^{2}+(15)^{2}}[/tex]
[tex]r=\sqrt{289}[/tex]
[tex]r=17\ units[/tex]
step 2
Find the equation of the circle
The equation of the circle in standard form is equal to
[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]
where
(h,k) is the center
r is the radius
substitute
[tex](x+7)^{2}+(y+1)^{2}=17^{2}[/tex]
[tex](x+7)^{2}+(y+1)^{2}=289[/tex]
step 3
Find the y-coordinate of the point (-15.y)
substitute the x-coordinate in the equation of the circle and solve for y
[tex](-15+7)^{2}+(y+1)^{2}=289[/tex]
[tex](-8)^{2}+(y+1)^{2}=289[/tex]
[tex]64+(y+1)^{2}=289[/tex]
[tex](y+1)^{2}=289-64[/tex]
[tex](y+1)^{2}=225[/tex]
square root both sides
[tex](y+1)=(+/-)15[/tex]
[tex]y=-1(+/-)15[/tex]
[tex]y1=-1(+)15=14[/tex]
[tex]y2=-1(-)15=-16[/tex]
therefore
The point (-15,14) and the point (-15,-16) lies on the circle
see the attached figure to better understand the problem
