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

Part 1) The radius of the circle is 17 units

Part 2) The point (-15,14) and (-15,-16) lies on this circle

Explanation:

step 1

Find the radius of the circle

we know that

The distance from a circle's center to any point on the circle is called the radius of the circle

the formula to calculate the distance between two points is equal to

`d=\sqrt{(y2-y1)^(2)+(x2-x1)^(2)}`

Find the distance (radius) between the points (-7,-1) and (8,7)

substitute in the formula

`r=\sqrt{(7+1)^(2)+(8+7)^(2)}`

`r=\sqrt{(8)^(2)+(15)^(2)}`

`r=√((289)\ units`

`r=17\ units`

step 2

Find the equation of the circle in center radius form

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

where

(h,k) is the center of the circle

r is the radius of the circle

we have

`(h,k)=(-7,-1)\\r=17\ units`

substitute

`(x+7)^2+(y+1)^2=17^2`

`(x+7)^2+(y+1)^2=289`

Remember that

If the point (-15,y) lie on the circle, then the ordered pair must satisfy the equation of the circle

substitute the value of x=-15 in the equation

`(-15+7)^2+(y+1)^2=289`

solve for y

`(-8)^2+(y+1)^2=289`

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

`(y+1)^2=225`

take square root both sides

`y+1=(+/-)15\\y=-1(+/-)15`

`y=-1+15=14\\y=-1-15=-16`

therefore

The point (-15,14) and (-15,-16) lies on this circle

see the attached figure to better understand the problem