Question

which circle(s) contain the point (1,4)? select all that apply. a. the circle centered at (4,0) with radius 5 b. the circle centered at (-2,8) with radius 5 c. the circle centered at (1,4) with radius 4 d. the circle centered at (1,0) with radius 4

Explanation:

Step1: Recall circle inclusion rule

A point $(x,y)$ is inside/on a circle with center $(h,k)$ and radius $r$ if $\sqrt{(x-h)^2+(y-k)^2} \leq r$, or equivalently $(x-h)^2+(y-k)^2 \leq r^2$.

Step2: Test Option A

Substitute $(1,4)$, $(4,0)$, $r=5$:
$\displaystyle(1-4)^2+(4-0)^2 = (-3)^2+4^2 = 9+16=25$
$25 = 5^2$, so the point is on the circle.

Step3: Test Option B

Substitute $(1,4)$, $(-2,8)$, $r=5$:
$\displaystyle(1-(-2))^2+(4-8)^2 = 3^2+(-4)^2 = 9+16=25$
$25 = 5^2$, so the point is on the circle.

Step4: Test Option C

Substitute $(1,4)$, $(1,4)$, $r=4$:
$\displaystyle(1-1)^2+(4-4)^2 = 0+0=0$
$0 \leq 4^2$, so the point is the center (inside the circle).

Step5: Test Option D

Substitute $(1,4)$, $(1,0)$, $r=4$:
$\displaystyle(1-1)^2+(4-0)^2 = 0+16=16$
$16 = 4^2$, so the point is on the circle.

Answer:

A. the circle centered at (4,0) with radius 5
B. the circle centered at (-2,8) with radius 5
C. the circle centered at (1,4) with radius 4
D. the circle centered at (1,0) with radius 4