Question

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.

Answer 1

well, we know the center of the circle is at (-7, -1), and we also know a point on it is (8, 7), hmmm what's the distance between those points anyway?

well, a distance from a point on the circle and the center is namely the definition of its radius.

`\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\(\stackrel{x_1}{-7}~,~\stackrel{y_1}{-1})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{7})\qquad \qquad d = √(( x_2- x_1)^2 + ( y_2- y_1)^2)\\\\\\\stackrel{radius}{r}=√([8-(-7)]^2+[7-(-1)]^2)\implies r=√((8+7)^2+(7+1)^2)\\\\\\r=√(15^2+8^2)\implies r=√(289)\implies r=17`

so, we know the distance from a point to the center.... ok, so the point (-15, y) is on the circle too, therefore it must have the same distance "r" too,

`\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\\stackrel{center}{(\stackrel{x_1}{-7}~,~\stackrel{y_1}{-1})}\qquad (\stackrel{x_2}{-15}~,~\stackrel{y_2}{y})\\\\\\\stackrel{r}{17}=√([-15-(-7)]^2+[y-(-1)]^2)\\\\\\17=√((-15+7)^2+(y+1)^2)\implies 17^2=(-15+7)^2+(y+1)^2\\\\\\289=(-8)^2+(y+1)^2\implies 289=64+(y+1)^2\\\\\\225=(y+1)^2\implies √(225)=y+1\implies 15=y+1\implies 14=y`

Answer 2

`(x+7)^ +(y+1)^2 = r^2`

We know a point given who lies on the circel (x =8, y=7) and if we replace we got:

`(8+7)^2 + (7+1)^2 = r^2`

`289 = r^2`

And if we take the square root we got:

`r = √(289)= 17`

`(-15+7)^2 +(y+1)^2 = 17^2`

And if we simplify we got:

`64 +(y+1)^2 = 289`

We can subtract 64 on both sides and we got:

`(y+1)^2 = 225`

And we can apply square roots on both side and we got:

`y+1= \pm √(225) = \pm 15`

And solving for y we got:

`y = 15-1 = 14`

`y = -15-1 = -16`

Explanation:

For this case we can use the general formula for a circle given by:

`(x-h)^2 +(y-k)^2 = r^2`

Where
`(h,k)` represent the center and r the radius. And for this special case we know this:
`(h = -7, k=-1)` and if we replace we got:

`(x-(-7))^2 +(y-(-1))^2 = r^2`

And if we simplify we got:

`(x+7)^ +(y+1)^2 = r^2`

We know a point given who lies on the circel (x =8, y=7) and if we replace we got:

`(8+7)^2 + (7+1)^2 = r^2`

`289 = r^2`

And if we take the square root we got:

`r = √(289)= 17`

For the other part of the problem we know that x = -15 and we need to find the coordinate of y, for a point who lies on the circle, and we can do this:

`(-15+7)^2 +(y+1)^2 = 17^2`

And if we simplify we got:

`64 +(y+1)^2 = 289`

We can subtract 64 on both sides and we got:

`(y+1)^2 = 225`

And we can apply square roots on both side and we got:

`y+1= \pm √(225) = \pm 15`

And solving for y we got:

`y = 15-1 = 14`

`y = -15-1 = -16`