Question

1. a circle has a center at (5, 7) and a radius of 5. which two of the following points are on the circle? (0, 12) (-5, -7) (5, 12) (10, 7) (10, 0) (5, 7)

Explanation:

Step1: Write circle equation

The circle with center $(h,k)=(5,7)$ and radius $r=5$ has the equation:
$$(x-5)^2 + (y-7)^2 = 5^2 = 25$$

Step2: Check point $(0,12)$

Substitute $(0,12)$ into the equation:

$$(0-5)^2 + (12-7)^2 = (-5)^2 + 5^2 = 25 + 25 = 50 eq 25$$

Not on the circle.

Step3: Check point $(-5,-7)$

Substitute $(-5,-7)$ into the equation:

$$(-5-5)^2 + (-7-7)^2 = (-10)^2 + (-14)^2 = 100 + 196 = 296 eq 25$$

Not on the circle.

Step4: Check point $(5,12)$

Substitute $(5,12)$ into the equation:
$$(5-5)^2 + (12-7)^2 = 0^2 + 5^2 = 0 + 25 = 25$$
On the circle.

Step5: Check point $(10,7)$

Substitute $(10,7)$ into the equation:
$$(10-5)^2 + (7-7)^2 = 5^2 + 0^2 = 25 + 0 = 25$$
On the circle.

Step6: Check point $(5,7)$

Substitute $(5,7)$ into the equation:

$$(5-5)^2 + (7-7)^2 = 0 + 0 = 0 eq 25$$

Not on the circle.

Step7: Check point $(10,0)$

Substitute $(10,0)$ into the equation:

$$(10-5)^2 + (0-7)^2 = 5^2 + (-7)^2 = 25 + 49 = 74 eq 25$$

Not on the circle.

Answer:

$(5,12)$ and $(10,7)$